NCERT exercise solutions of this chapter is helpful to improve the hold of the students in the problems related to sequences and series. All the questions of this exercise have been solved by subject experts.The Exercise 9.3 of NCERT Solutions for Class 11 Maths Chapter 9- Sequences and Series is based on the following topics:

- Geometric Progression (G. P.)
- General term of a G.P
- Sum to n terms of a G.P
- Geometric Mean (G.M.)

- Relationship Between A.M. and G.M.

These solutions are prepared by subject matter experts at BYJUâ€™S, describing the complete method of solving problems. By understanding the concepts used in NCERT Solutions for Class 11 Maths, students will be able to clear all their doubts related to the chapters of class 11.

## Download PDF of NCERT Solutions for Class 11 Maths Chapter 9- Sequences and Series Exercise 9.3

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### Access other exercise solutions of Class 11 Maths Chapter 9- Sequences and Series

Exercise 9.1 Solutions 14 Questions

Exercise 9.2 Solutions 18 Questions

Exercise 9.4 Solutions 10 Questions

Miscellaneous Exercise On Chapter 9 Solutions 32 Questions

#### Access Solutions for Class 11 Maths Chapter 9.3 Exercise

**1. Find the 20 ^{th}Â andÂ n^{th}terms of the G.P. 5/2, 5/4, 5/8, â€¦â€¦â€¦**

**Solution:**

Given G.P. isÂ 5/2, 5/4, 5/8, â€¦â€¦â€¦

Here,Â *a*Â = First term =Â 5/2

*r*Â = Common ratio =Â (5/4)/(5/2) = Â½

Thus, the 20^{th} term and n^{th} term

**2. Find the 12 ^{th}Â term of a G.P. whose 8^{th}Â term is 192 and the common ratio is 2.**

**Solution:**

Given,

The common ratio of the G.P.,Â *r*Â = 2

And, letÂ *a*Â be the first term of the G.P.

Now,

*a*_{8}Â =Â *ar*Â ^{8â€“1}Â =Â *ar*^{7}

*ar*^{7}Â = 192

*a*(2)^{7}Â = 192

*a*(2)^{7}Â = (2)^{6}Â (3)

**3. The 5 ^{th}, 8^{th}Â and 11^{th}Â terms of a G.P. areÂ p,Â qÂ andÂ s, respectively. Show thatÂ q^{2}Â =Â ps.**

**Solution:**

Letâ€™s takeÂ *a*Â to be the first term and*Â r*Â to be the common ratio of the G.P.

Then according to the question, we have

*a*_{5}Â =Â *a*Â *r*^{5â€“1Â }=Â *a*Â r^{4}Â =Â *p*Â â€¦ (i)

*a*_{8Â }=Â *a*Â *r*^{8â€“1Â }=Â *a*Â *r*^{7}Â =Â *q*Â â€¦ (ii)

*a*_{11}Â = aÂ *r*^{11â€“1Â }=Â *a*Â *r*^{10}Â =Â *sÂ *â€¦ (iii)

Dividing equation (ii) by (i), we get

**4. The 4 ^{th}Â term of a G.P. is square of its second term, and the first term is â€“3. Determine its 7^{th}Â term.**

**Solution:**

Letâ€™s considerÂ *a*Â to be the first term andÂ *r*Â to be the common ratio of the G.P.

Given, *a*Â = â€“3

And we know that,Â

*a _{n}*Â =Â

*ar*

^{n}^{â€“1}

So, *a*_{4Â }=Â *ar*^{3}Â = (â€“3)Â *r*^{3}

*a*_{2}Â =Â *a r*^{1}Â = (â€“3)Â *r*

Then from the question, we have

(â€“3)Â *r*^{3}Â = [(â€“3)Â *r*]^{2}

â‡’ â€“3*r*^{3}Â = 9Â *r*^{2}

â‡’Â *r*Â = â€“3

*a*_{7}Â =Â *a*Â *r*Â ^{7â€“1Â }=Â *a*Â *r*^{6}Â = (â€“3) (â€“3)^{6}Â = â€“ (3)^{7}Â = â€“2187

Therefore, the seventh term of the G.P. is â€“2187.

**5. Which term of the following sequences:**

**(a) 2, 2âˆš2, 4,â€¦ is 128 ? (b) âˆš3, 3, 3âˆš3,â€¦ is 729 ?**

**(c) 1/3, 1/9, 1/27, â€¦ is 1/19683 ?**

**Solution: **

(a) The given sequence, 2, 2âˆš2, 4,â€¦

We have,

a = 2 and r = 2âˆš2/2 = âˆš2

Taking the n^{th} term of this sequence as 128, we have

Therefore, the 13^{th} term of the given sequence is 128.

(ii) Given sequence, âˆš3, 3, 3âˆš3,â€¦

We have,

a = âˆš3 and r = 3/âˆš3 = âˆš3

Taking the n^{th} term of this sequence to be 729, we have

Therefore, the 12^{th} term of the given sequence is 729.

(iii) Given sequence, 1/3, 1/9, 1/27, â€¦

a = 1/3 and r = (1/9)/(1/3) = 1/3

Taking the n^{th} term of this sequence to be 1/19683, we have

Therefore, the 9^{th} term of the given sequence is 1/19683.

**6. For what values ofÂ x,Â the numbersÂ -2/7, x, -7/2 are in G.P?**

**Solution:**

The given numbers are -2/7, x, -7/2.

Common ratioÂ = x/(-2/7) = -7x/2Â

Also, common ratio =Â (-7/2)/x = -7/2x

Therefore, for*Â x*Â = Â± 1, the given numbers will be in G.P.

**7. Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 â€¦**

**Solution:**

Given G.P., 0.15, 0.015, 0.00015, â€¦

Here,Â *a*Â = 0.15 and r = 0.015/0.15 = 0.1**Â **

**8. Find the sum toÂ nÂ terms in the geometric progression âˆš7, âˆš21, 3âˆš7, â€¦.**

**Solution: **

The given G.P is âˆš7, âˆš21, 3âˆš7, â€¦.

Here,

a = âˆš7 and

**9. Find the sum toÂ nÂ terms in the geometric progression 1, -a, a^{2}, -a^{3} â€¦. (if a â‰ -1)**

**Solution:**

The given G.P. isÂ 1, -a, a^{2}, -a^{3} â€¦.

Here, the first term =Â *a*_{1}Â = 1

And the common ratio =Â *r*Â = â€“Â *a*

We know that,

**10. Find the sum toÂ nÂ terms in the geometric progression x^{3}, x^{5}, x^{7}, â€¦ (if x â‰ Â±1 )**

**Solution:**

Given G.P. isÂ x^{3}, x^{5}, x^{7}, â€¦

Here, we haveÂ *a*Â =Â *x*^{3}Â andÂ *r*Â =Â *x*^{5}/x^{3} = x^{2}

**11. Evaluate: **

**Solution: **

**12. The sum of first three terms of a G.P. isÂ 39/10 and their product is 1. Find the common ratio and the terms.**

**Solution:**

LetÂ a/r, a, ar be the first three terms of the G.P.

a/r + a + ar = 39/10 â€¦â€¦ (1)

(a/r) (a) (ar) = 1 â€¦â€¦.. (2)

From (2), we have

a^{3} = 1

Hence, a = 1 [Considering real roots only]

Substituting the value of a in (1), we get

1/r + 1 + r = 39/10

(1 + r + r^{2})/r = 39/10

10 + 10r + 10r^{2} = 39r

10r^{2} â€“ 29r + 10 = 0

10r^{2} â€“ 25r â€“ 4r + 10 = 0

5r(2r â€“ 5) â€“ 2(2r â€“ 5) = 0

(5r â€“ 2) (2r â€“ 5) = 0

Thus,

r = 2/5 or 5/2

Therefore, the three terms of the G.P. are 5/2, 1 and 2/5.

**13. How many terms of G.P. 3, 3 ^{2}, 3^{3}, â€¦ are needed to give the sum 120?**

**Solution: **

Given G.P. is 3, 3^{2}, 3^{3}, â€¦

Letâ€™s consider thatÂ *n*Â terms of this G.P. be required to obtain the sum of 120.

We know that,

Here,Â *a*Â = 3 andÂ *r*Â = 3

Equating the exponents we get, *n*Â = 4

Therefore, four terms of the given G.P. are required to obtain the sum as 120.

**14. The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to Â nÂ terms of the G.P.**

**Solution: **

Letâ€™s assume the G.P. to beÂ *a*,Â *ar*,Â *ar*^{2},Â *ar*^{3}, â€¦

Then according to the question, we have

*aÂ *+Â *ar*Â +Â *ar*^{2}Â = 16 andÂ *ar*^{3Â }+Â *ar*^{4}Â +Â *ar*^{5Â }= 128

*a*Â (1 +Â *r*Â +Â *r*^{2}) = 16 â€¦ (1) and,

*ar*^{3}(1 +Â *r*Â +Â *r*^{2}) = 128 â€¦ (2)

Dividing equation (2) by (1), we get

r^{3} = 8

r = 2

Now, using r = 2 in (1), we get

a (1 + 2 + 4) = 16

a (7) = 16

a = 16/7

Now, the sum of terms is given as

**15. Given a G.P. withÂ aÂ = 729 and 7^{th}Â term 64, determine S_{7}.**

**Solution: **

Given,

*a*Â = 729 and *a*_{7}Â = 64

LetÂ *r*Â be the common ratio of the G.P.

Then we know that,Â *a _{n}*Â =Â

*a r*

^{n}^{â€“1}

*a*_{7}Â =Â *ar*^{7â€“1}Â = (729)*r*^{6}

â‡’ 64 = 729Â *r*^{6}

*r*^{6} = 64/729

*r*^{6} = (2/3)^{6}

r = 2/3

And, we know that

= 3^{7} â€“ 2^{7}

= 2187 â€“ 128

= 2059

**16. Find a G.P. for which sum of the first two terms is â€“4 and the fifth term is 4 times the third term.**

**Solution: **

ConsiderÂ *a*Â to be the first term andÂ *r*Â to be the common ratio of the G.P.

Given, S_{2} = -4

Then, from the question we have

And,

a_{5} = 4 x a_{3}

ar^{4} = 4ar^{2}

r^{2} = 4

r = Â± 2

Using the value of r in (1), we have

Therefore, the required G.P is

-4/3, -8/3, -16/3, â€¦. Or 4, -8, 16, -32, â€¦â€¦

**17. If the 4 ^{th}, 10^{th}Â and 16^{th}Â terms of a G.P. areÂ x, yÂ andÂ z, respectively. Prove thatÂ x,Â y,Â zÂ are in G.P.**

**Solution:**

LetÂ *a*Â be the first term andÂ *r*Â be the common ratio of the G.P.

According to the given condition,

*a*_{4}Â =Â *a*Â *r*^{3}Â =Â *x*Â â€¦ (1)

*a*_{10}Â =Â *a*Â *r*^{9}Â =*Â y*Â â€¦ (2)

*a*_{16}^{Â }=*Â a r*^{15Â }=Â *z*Â â€¦ (3)

On dividing (2) by (1), we get

**18. Find the sum to Â nÂ terms of the sequence, 8, 88, 888, 8888â€¦**

**Solution:**

Given sequence: 8, 88, 888, 8888â€¦

This sequence is not a G.P.

But, it can be changed to G.P. by writing the terms as

*S _{n}*Â = 8 + 88 + 888 + 8888 + â€¦â€¦â€¦â€¦â€¦.. toÂ

*n*Â terms

**19. Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2,Â 1/2.**

**Solution:**

The required sum =Â 2 x 128 + 4 x 32 + 8 x 8 + 16 x 2 + 32 x Â½

= 64[4 + 2 + 1 + Â½ + 1/2^{2}]

Now, itâ€™s seen that

4, 2, 1,Â Â½, 1/2^{2} is a G.P.

With first term,Â *a*Â = 4

Common ratio,Â *r*Â =1/2

We know,

Â

Therefore, the required sum =Â 64(31/4) = (16)(31) = 496

**20. Show that the products of the corresponding terms of the sequencesÂ a, ar, ar ^{2}, â€¦ar^{n-1} and A, AR, AR^{2}, â€¦ AR^{n-1}Â form a G.P, and find the common ratio.**

**Solution:**

To be proved: The sequence,Â *aA*,Â *arAR*,Â *ar*^{2}*AR*^{2}, â€¦*ar ^{n}*

^{â€“1}

*AR*

^{n}^{â€“1}, forms a G.P.

Now, we have

Therefore, the above sequence forms a G.P. and the common ratio isÂ *rR*.

**21. Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4 ^{th}Â by 18.**

**Solution: **

ConsiderÂ *a*Â to be the_{Â }first term andÂ *r*Â to be the common ratio of the G.P.

Then,

*a*_{1}Â =Â *a*,Â *a*_{2}Â =Â *ar*,Â *a*_{3}Â =Â *ar*^{2},Â *a*_{4}Â =Â *ar*^{3}

From the question, we have

*a*_{3}Â =Â *a*_{1}Â + 9

*ar*^{2}Â =Â *a*Â + 9 â€¦ (i)

*a*_{2}Â =Â *a*_{4}Â + 18

*arÂ *=Â *ar*^{3}Â + 18 â€¦ (ii)

So, from (1) and (2), we get

*a*(*r*^{2}Â â€“ 1) = 9 â€¦ (iii)

*arÂ *(1â€“Â *r*^{2}) = 18 â€¦ (iv)

Now, dividing (4) by (3), we get

-r = 2

r = -2

On substituting the value ofÂ *r*Â in (i), we get

4*aÂ *=Â *a*Â + 9

3*a*Â = 9

âˆ´Â *a*Â = 3

Therefore, the first four numbers of the G.P. are 3, 3(â€“ 2), 3(â€“2)^{2}, and 3(â€“2)^{3}Â

i.e., 3Â¸â€“6, 12, and â€“24.

**22. If the p ^{th}, q^{th} and r^{th}Â terms of a G.P. areÂ a, bÂ andÂ c, respectively. Prove that a^{q-r }b^{r-p }c^{p-q} = 1**

**Solution: **

Letâ€™s take *A*Â to be the first term andÂ *R*Â to be the common ratio of the G.P.

Then according to the question, we have

*AR ^{p}*

^{â€“1Â }=Â

*a*

*AR ^{q}*

^{â€“1Â }=Â

*b*

*AR ^{r}*

^{â€“1Â }=Â

*c*

Then,

*a ^{qâ€“r}*

^{Â }

*b*

^{râ€“p}^{Â }

*c*

^{pâ€“q}=Â *A ^{q}*

^{â€“}

*Ã—Â*

^{rÂ }*R*

^{(}

^{p}^{â€“1) (qâ€“r)}Â Ã— A

^{r}^{â€“}

*Â Ã—Â*

^{p}*R*

^{(}

^{q}^{â€“1) (}

^{r}^{â€“}

^{p}^{)}Â Ã—Â

*A*

^{p}^{â€“}

*Â Ã—Â*

^{q}*R*

^{(}

^{rÂ }^{â€“1)(}

^{p}^{â€“}

^{q}^{)}

=Â *Aq*^{Â â€“Â }^{r}^{Â +Â }^{r}^{Â â€“Â }^{p}^{Â +Â }^{p}^{Â â€“Â }* ^{q}*Â Ã—Â

*R*Â

^{(}

^{pr}^{Â â€“Â }

^{pr}^{Â â€“Â }

^{q}^{Â +Â }

^{r}^{) + (}

^{rq}^{Â â€“}

^{Â rÂ }^{+Â }

^{p}^{Â â€“Â }

^{pq}^{) + (}

^{pr}^{Â â€“Â }

^{p}^{Â â€“Â }

^{qr}^{Â +Â }

^{q}^{)}

=Â *A*^{0}Â Ã—Â *R*^{0}

= 1

Hence proved.

**23. If the first and theÂ n^{th}Â term of a G.P. areÂ aÂ adÂ b, respectively, and ifÂ PÂ is the product ofÂ nÂ terms, prove thatÂ P^{2}Â = (ab)^{n}.**

**Solution:**

Given, the first term of the G.P isÂ *a*Â and the last term isÂ *b*.

Thus,

The G.P. isÂ *a*,Â *ar*,Â *ar*^{2},Â *ar*^{3}, â€¦Â *ar ^{n}*

^{â€“1}, whereÂ

*r*Â is the common ratio.

Then,

*b*Â =Â *ar ^{n}*

^{â€“1}Â â€¦ (1)

*P*Â = Product ofÂ *n*Â terms

= (*a*) (*ar*) (*ar*^{2}) â€¦ (*ar ^{n}*

^{â€“1})

= (*a*Â Ã—Â *a*Â Ã—â€¦*a*) (*r*Â Ã—Â *r*^{2}Â Ã— â€¦*r ^{n}*

^{â€“1})

=Â *a ^{n}*

^{Â }

*r*Â

^{1 + 2 +â€¦(}

^{n}^{â€“1)}Â â€¦ (2)

Here, 1, 2, â€¦(*n*Â â€“ 1) is an A.P.

And, the product of n terms P is given by,

**24. Show that the ratio of the sum of firstÂ nÂ terms of a G.P. to the sum of terms fromÂ .**

**Solution:**

LetÂ *a*Â be the first term andÂ *rÂ *be the common ratio of the G.P.

Since there areÂ *n*Â terms from (*n*Â +1)^{th}Â to (2*n*)^{th}Â term,

Sum of terms from(*n*Â + 1)^{th}Â to (2*n*)^{th}Â termÂ

*a*Â ^{nÂ }^{+1}Â =Â *ar*Â ^{n + 1}Â ^{â€“ 1}Â =Â *ar ^{n}*

Thus, required ratio =Â

Thus, the ratio of the sum of firstÂ *n*Â terms of a G.P. to the sum of terms from (*n*Â + 1)^{th}Â to (2*n*)^{thÂ }term is^{Â .}

**25. IfÂ a, b, cÂ andÂ dÂ are in G.P. show thatÂ (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d^{2}) = (ab + bc + cd)^{2}.**

**Solution: **

Given, *a*,Â *b*,Â *c*,Â *d*Â are in G.P.

So, we have

*bc*Â =Â *ad*Â â€¦ (1)

*b*^{2}Â =Â *acÂ *â€¦ (2)

*c*^{2}Â =Â *bd*Â â€¦ (3)

Taking the R.H.S. we have

R.H.S.

= (*ab*Â +Â *bc*Â +Â *cd*)^{2}

= (*ab*Â +Â *adÂ *+Â *cd*)^{2}Â [Using (1)]

= [*ab*Â +Â *d*Â (*a*Â +Â *c*)]^{2}

=Â *a*^{2}*b*^{2}Â + 2*abd*Â (*a*Â +Â *c*) +Â *d*^{2}Â (*a*Â +Â *c*)^{2}

=Â *a*^{2}*b*^{2}Â +2*a*^{2}*bd*Â + 2*acbd*Â +Â *d*^{2}(*a*^{2}Â + 2*ac*Â +Â *c*^{2})

=Â *a*^{2}*b*^{2}Â + 2*a*^{2}*c*^{2}Â + 2*b*^{2}*c*^{2}Â +Â *d*^{2}*a*^{2}Â + 2*d*^{2}*b*^{2}Â +Â *d*^{2}*c*^{2}Â [Using (1) and (2)]

=Â *a*^{2}*b*^{2}Â +Â *a*^{2}*c*^{2}Â +Â *a*^{2}*c*^{2}Â +Â *b*^{2}*c*^{2Â }+Â *b*^{2}*c*^{2}Â +Â *d*^{2}*a*^{2}Â +Â *d*^{2}*b*^{2}Â +Â *d*^{2}*b*^{2}Â +Â *d*^{2}*c*^{2}

=Â *a*^{2}*b*^{2}Â +Â *a*^{2}*c*^{2}Â +Â *a*^{2}*d*^{2Â }+Â *b*^{2Â }Ã—Â *b*^{2}Â +Â *b*^{2}*c*^{2}Â +Â *b*^{2}*d*^{2}Â +Â *c*^{2}*b*^{2}Â +Â *c*^{2Â }Ã—Â *c*^{2}Â +Â *c*^{2}*d*^{2}

=Â *a*^{2}(*b*^{2}Â +Â *c*^{2}Â +Â *d*^{2}) +Â *b*^{2}Â (*b*^{2}Â +Â *c*^{2}Â +Â *d*^{2}) +Â *c*^{2}Â (*b*^{2}+Â *c*^{2}Â +Â *d*^{2})

= (*a*^{2}Â +Â *b*^{2}Â +Â *c*^{2}) (*b*^{2}Â +Â *c*^{2}Â +Â *d*^{2})

= L.H.S.

Thus, L.H.S. = R.H.S.

Therefore,

(a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d^{2}) = (ab + bc + cd)^{2}

**26. Insert two numbers between 3 and 81 so that the resulting sequence is G.P.**

**Solution:**

Letâ€™s assumeÂ *G*_{1}Â andÂ *G*_{2}Â to be two numbers between 3 and 81 such that the series 3,Â *G*_{1},Â *G*_{2}, 81 forms a G.P.

And letÂ *a*Â be the first term andÂ *r*Â be the common ratio of the G.P.

Now, we have the 1^{st} term as 3 and the 4^{th} term as 81.

81 = (3)Â *(r*)^{3}

*r*^{3}Â = 27

âˆ´Â *r*Â = 3 (Taking real roots only)

ForÂ *r*Â = 3,

*G*_{1}Â =Â *ar*Â = (3) (3) = 9

*G*_{2}Â =Â *ar*^{2}Â = (3) (3)^{2}Â = 27

Therefore, the two numbers which can be inserted between 3 and 81 so that the resulting sequence becomes a G.P are 9 and 27.

**27. Find the value of Â nÂ so thatÂ may be the geometric mean betweenÂ aÂ andÂ b.**

**Solution: **

We know that,

The G. M. ofÂ *a*Â andÂ *b*Â isÂ given by âˆšab.

Then from the question, we have

Â

By squaring both sides, we get

**28. The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio.**

**Solution:**

Consider the two numbers beÂ *a*Â andÂ *b*.

Then, G.M. =Â âˆšab.

From the question, we have

**29. If A and G be A.M. and G.M., respectively between two positive numbers, prove that the **

**numbers are . **

**Solution:**

Given that*Â A*Â andÂ *G*Â are A.M. and G.M. between two positive numbers.

And, let these two positive numbers beÂ *a*Â andÂ *b*.

**30. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2 ^{nd}Â hour, 4^{th}Â hour andÂ n^{th}Â hour?**

**Solution:**

Given, the number of bacteria doubles every hour. Hence, the number of bacteria after every hour will form a G.P.

Here we have,Â *a*Â = 30 andÂ *r*Â = 2

So,Â *a*_{3}Â =Â *ar*^{2}Â = (30) (2)^{2}Â = 120

Thus, the number of bacteria at the end of 2^{nd}Â hour will be 120.

And, *a*_{5}Â =Â *ar*^{4}Â = (30) (2)^{4}Â = 480

The number of bacteria at the end of 4^{th}Â hour will be 480.

*a _{n}*

_{Â +1Â }=Â

*ar*Â = (30) 2

^{n}

^{n}Therefore, the number of bacteria at the end ofÂ *n*^{th}Â hour will be 30(2)* ^{n}*.

**31. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?**

**Solution:**

Given,

The amount deposited in the bank is Rs 500.

At the end of first year, amount = Rs 500(1 + 1/10)Â = Rs 500 (1.1)

At the end of 2^{nd}Â year, amount = Rs 500 (1.1) (1.1)

At the end of 3^{rd}Â year, amount = Rs 500 (1.1) (1.1) (1.1) and so onâ€¦.

Therefore,

The amount at the end of 10 years = Rs 500 (1.1) (1.1) â€¦ (10 times)

= Rs 500(1.1)^{10}

**32. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.**

**Solution:**

Letâ€™s consider the roots of the quadratic equation to beÂ *a*Â andÂ *b*.

Then, we have

We know that,

A quadratic equation can be formed as,

*x*^{2 }â€“Â *x*Â (Sum of roots) + (Product of roots) = 0

*x*^{2}Â â€“Â *x*Â (*a*Â +Â *b*) + (*ab*) = 0

*x*^{2}Â â€“ 16*x*Â + 25 = 0 [Using (1) and (2)]

Therefore, the required quadratic equation isÂ *x*^{2}Â â€“ 16*x*Â + 25 = 0