JEE Main 2024 Question Paper Solution Discussion Live JEE Main 2024 Question Paper Solution Discussion Live

Arithmetic Progression Solved Examples

In series and sequence, arithmetic, geometric and harmonic sequences are important concepts. A set of numbers which follows a certain pattern is called a sequence; for example, 3, 6, 9, 12, โ€ฆโ€ฆโ€ฆ.(a sequence of multiple of 3), also even numbers, odd numbers, follow a particular pattern. The numbers are named as terms of a sequence.ย  In this article, we come across arithmetic progression solved examples and its definition.

Arithmetic Progression Definition

A sequence of numbers in which the difference between the two consecutive numbers is always constant is termed an arithmetic progression. The difference is called the common difference.

  • The first term of an AP is denoted as โ€œaโ€.
  • The common difference is represented as โ€œdโ€.
  • The common difference can either be positive or negative.

Arithmetic Progression Sequence

A sequence containing finite terms is called a finite sequence, whereas a sequence with infinite terms, that is, without the last term, can be defined as an infinite sequence.

Properties of an Arithmetic Progression

1. If we add a constant to each term of an A.P., the resulting sequence will also be an A.P.

2. If we subtract a constant from each term of an A.P., the resulting sequence will also be an A.P.

3. If we multiply a constant by each term of an A.P., the resulting sequence will also be an A.P.

4. If each term of an A.P. is divided by a non-zero constant, the resulting sequence will also be an A.P.

We use the following notations for an A.P.:

a= First term

d = Common difference

l = Last term

n = No. of terms

Sn = Sum of n terms of an A.P

Arithmetic Progression Formula

Formula Terms
The general formula for finding the nth term of an AP.
\(\begin{array}{l}T_n = a + (n-1)d\end{array} \)
Where d = an – an-1 and Tn is the nth term of the sequence.
The formula for finding the sum to n terms of an AP.
\(\begin{array}{l}S_n = \frac{n}{2} [2a + (n-1)d]\end{array} \)
Where Sn is the sum of n terms, a is the first term, and d is the common difference.
When the last term is known, then the sum of n terms.
\(\begin{array}{l}S_n = \frac{n}{2} [a + l]\end{array} \)

Also Read:

Solved Examples on Arithmetic Progression

Example 1: Let the bth term of an A.P. be q, and the cth term be p, then find its dth term.

Solution:

Tb = a + (b โˆ’ 1)d = q …..(i) and

Tc = a + (c โˆ’ 1)d = p โ€ฆ… (ii)

From (i) and (ii),

We get d = โˆ’(bโˆ’c)/(bโˆ’c) = โˆ’1

Putting the value of d in equation (i), then a = b + c โˆ’ 1

Now, the dth term is given by A.P.

Td = a + (d โˆ’ 1)d

= (b + c โˆ’ 1) + (d โˆ’ 1)(โˆ’1)

= b + c โˆ’ d

Example 2: Write the sum of the numbers which are divisible by 8 between 100 and 1000.

Solution:

The series will be 104, 112, โ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆ.., 992

Common difference = 8

n = 992/8 – 96/8 = 124 – 12 = 112

Hence, the required sum is,

Sn = 112/2 (104 + 992) = 61376

Example 3: If the 8th term of an AP is zero, then what is the ratio of its 28th and 18th terms?

Solution:

Given that 8th term = a+(8โˆ’1)d = 0

โ‡’ a+7d = 0

Now, the ratio of 28th and 18th terms

(a + 27d) /(a + 17d) = {(a + 7d) + 20d} / {(a + 7d) + 10d} = 20d/10d = 2/1

Example 4: If tan (mฮธ) = tan (nฮธ), in which sequence will the different values of ฮธ be?

Solution:

We have tan (mฮธ) = tan(nฮธ)

โ‡’ mฮธ = Mฯ€ + (nฮธ)

โ‡’ ฮธ = Mฯ€m โˆ’ n, putting M = 1, 2, 3………, we get

ฯ€/mโˆ’n, 2ฯ€/mโˆ’n, 3ฯ€/mโˆ’n,……… which are obviously in A.P.

Since common difference d = ฯ€/m โˆ’ n

Example 5: What will the nth term of the series 3.9 + 6.12 + 9.15 + 12.18 + …..be?

Solution:

Given series 3.9+6.12+9.15+12 .18+…..

The first factors are 3, 6, 9, and 12, whose nth term is 3n, and the second factors are 9, 12, 15, and 18.

tn = [9 + (n โˆ’ 1)3] = (3n + 6)

Hence, the nth term of the given series = 3n(3n + 6)

Example 6: If 3x, x + 9, 6x + 1 are in A.P., what will the value of x be?

Solution:

3x, x + 9, 6x + 1 are in A.P.

Therefore, (x + 9) = (3x) + (6x + 1)/2 =ย  9x + 1 / 2

โ‡’ 2x + 18 = 9x + 1

โ‡’ 7x = 17

โ‡’ x = 2.42

Example 7: Find the sum of all natural numbers lying between 101 and 199, which are multiples of 5.

Solution:

We know the nearest number to 101, which is a multiple of 5 and lies between 101 and 199, is 105. Similarly, 195 is a multiple of 5, which lies between 101 and 199. So, we take the last term of A.P. as 195.

We have the first term, a = 105ย 

The common difference, d = 5

The last term, l = 195

l = a + (n – 1) d .

195 = 105 + (n – 1)5

195 – 105 = (n – 1)5

5n – 5 = 90

5n = 90 + 5 = 95

n = 95/5 = 19

Sum of n terms = (n/2) (a + l)

= (19/2) (105 + 195) = 2850

Hence, the required sum is 2850.

Example 8: At what sum of all two-digit numbers, which, when divided by 4, yield unity as a reminder?

Solution:

The given numbers are 13, 17, ….. 97.

This is an AP with first term 13 and common difference 4.

Let the number of terms be n.

Then, 97 = 13 + (n โˆ’ 1)4

โ‡’ 4n = 88
โ‡’ n = 22

Therefore, the sum of the numbers = 222[13 + 97] = 11(110) = 1210.

Example 9:ย If a1, a2, a3,…, an are in A.P. with a common difference, d, then the sum of the following series

\(\begin{array}{l}\sin d(\cos \text{ec}\,{{a}_{1}}.co\text{sec}\,{{a}_{2}}+\text{cosec}\,{{a}_{2}}.\text{cosec}\,{{a}_{3}}+………..+\text{cosec}\ {{a}_{n-1}}\text{cosec}\ {{a}_{n}})\end{array} \)
is

Solution:

As given

\(\begin{array}{l}d={{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}=….={{a}_{n}}-{{a}_{n-1}} \\\sin d\,\{co\text{sec}\ {{a}_{1}}co\text{sec}\ {{a}_{2}}+…..+\text{cosec}\ {{a}_{n-1}}\text{cosec}\ {{a}_{n}}\} \\=\frac{\sin ({{a}_{2}}-{{a}_{1}})}{\sin {{a}_{1}}.\ \sin {{a}_{2}}}+……+\frac{\sin ({{a}_{n}}-{{a}_{n-1}})}{\sin {{a}_{n-1}}\sin {{a}_{n}}} \\=(\cot {{a}_{1}}-\cot {{a}_{2}})+(\cot {{a}_{2}}-\cot {{a}_{3}})+….+(\cot {{a}_{n-1}}-\cot {{a}_{n}}) \\=\cot {{a}_{1}}-\cot {{a}_{n}}.\\\end{array} \)

Example 10:ย If a1, a2, a3,…, an are in A.P., where ai > 0 for all i, then the value of

\(\begin{array}{l}\frac{1}{\sqrt{{{a}_{1}}}+\sqrt{{{a}_{2}}}}+\frac{1}{\sqrt{{{a}_{2}}}+\sqrt{{{a}_{3}}}}+ ……..+\frac{1}{\sqrt{{{a}_{n-1}}}+\sqrt{{{a}_{n}}}}=\end{array} \)

Solution:

As given

\(\begin{array}{l}{{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}=…….={{a}_{n}}-{{a}_{n-1}}=d\end{array} \)
where d is the common difference of the given A.P. Alsoย 
\(\begin{array}{l}{{a}_{n}}={{a}_{1}}+(n-1)d\end{array} \)
.ย Then by rationalising each term,

\(\begin{array}{l}\frac{1}{\sqrt{{{a}_{2}}}+\sqrt{{{a}_{1}}}}+\frac{1}{\sqrt{{{a}_{3}}}+\sqrt{{{a}_{2}}}}+….+\frac{1}{\sqrt{{{a}_{n}}}+\sqrt{{{a}_{n-1}}}}\\ =\frac{\sqrt{{{a}_{2}}}-\sqrt{{{a}_{1}}}}{{{a}_{2}}-{{a}_{1}}}+\frac{\sqrt{{{a}_{3}}}-\sqrt{{{a}_{2}}}}{{{a}_{3}}-{{a}_{2}}}+…..+\frac{\sqrt{{{a}_{n}}}-\sqrt{{{a}_{n-1}}}}{{{a}_{n}}-{{a}_{n-1}}}\\=\frac{1}{d}\left\{ \sqrt{{{a}_{2}}}-\sqrt{{{a}_{1}}}+\sqrt{{{a}_{3}}}-\sqrt{{{a}_{2}}}+……+\sqrt{{{a}_{n}}}-\sqrt{{{a}_{n-1}}} \right\}\\=\frac{1}{d}\left\{ \sqrt{{{a}_{n}}}-\sqrt{{{a}_{1}}} \right\}=\frac{1}{d}\left( \frac{{{a}_{n}}-{{a}_{1}}}{\sqrt{{{a}_{n}}}+\sqrt{{{a}_{1}}}} \right)\\=\frac{1}{d}\left\{ \frac{(n-1)d}{\sqrt{{{a}_{n}}}+\sqrt{{{a}_{1}}}} \right\}\\=\frac{n-1}{\sqrt{{{a}_{n}}}+\sqrt{{{a}_{1}}}}.\end{array} \)

Example 11:

\(\begin{array}{l}\text{If}\ 1,\,\,{{\log }_{9}}({{3}^{1-x}}+2),\,\,{{\log }_{3}}({{4.3}^{x}}-1)\ \text{are in A.P.},\end{array} \)
then x equals

Solution:

The given numbers are in A.P.

\(\begin{array}{l}2{{\log }_{9}}({{3}^{1-x}}+2)={{\log }_{3}}({{4.3}^{x}}-1)+1\\ 2{{\log }_{{{3}^{2}}}}({{3}^{1-x}}+2)={{\log }_{3}}({{4.3}^{x}}-1)+{{\log }_{3}}3 \\ \frac{2}{2}{{\log }_{3}}({{3}^{1-x}}+2)={{\log }_{3}}[3({{4.3}^{x}}-1) \\ {{3}^{1-x}}+2=3\,({{4.3}^{x}}-1) \\ \frac{3}{y}+2=12y-3, \text{where}\ y={{3}^{x}} \\ 12{{y}^{2}}-5y-3=0 \\y=\frac{-1}{3} \text{or}\ \frac{3}{4}\Rightarrow {{3}^{x}}=\frac{-1}{3}\,\, \\\text{or}\ {{3}^{x}}=\frac{3}{4} x={{\log }_{3}}\,(3/4) \\\Rightarrow x=1-{{\log }_{3}}4.\end{array} \)

Video Lessons

Sequence and Series

Arithmetic Progression & Its Properties

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