Sum of Infinite Terms

Q.

How many terms of G.P. 3, 3², 3³, … are needed to give the sum 120?

Q. Arg(z)+arg(conjugate of z) is equal to 0 prove this.

Q.

The 5^th, 8^th and 11^th terms of a G.P. are p, q and s, respectively. Show that q² = ps.

Q.

Given a G.P. with a = 729 and 7^th term 64, determine S₇.

Q. If $5+55+555+\cdots \text{to}n\text{terms}=\frac{5}{81}({10}^{n+1}+x+yn)$, then the value of $y-x$ is

1. $-19$
2. $-1$
3. $19$
4. $1$

Q. If three non-zero numbers are in A.P. and the squares of those numbers taken in same order are in G.P., then the common ratio can be

1. $0$
2. $1$
3. $3+2\sqrt{2}$
4. $3-2\sqrt{2}$

Q.

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

Q.

The distance between the planes 3x - 2y + 6z + 21 = 0 and - 6x + 4y - 12z + 35 = 0 is:

1. 

2. 

3. 

4. 2

Q.

If $S_1,S_2,S_3,\dots ,Sn$ are the sums of infinite G.P.s. whose first terms are $1,2,3,\dots ,n$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n+1}$ respectively, then ${\sum }_{i=1^n}{S}_i=$

1. $\frac{n\left(n+3\right)}{2}$

2. $\frac{n\left(n+4\right)}{2}$

3. $\frac{n\left(n-3\right)}{2}$

4. $\frac{n\left(n+1\right)}{2}$

Q.

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 hour, 4^th hour and n^th hour?

Q.

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from .

Q. The determinant $\left|\begin{array}{ccc}{b}^2-ab& b-c& bc-ac\\ ab-a^2& a-b& b^2-ab\\ bc-ca& c-a& ab-a^2\end{array}\right|$ equals

(a) $abc\left(b-c\right)\left(c-a\right)\left(a-b\right)$
(b) $\left(b-c\right)\left(c-a\right)\left(a-b\right)$
(c) $\left(a+b+c\right)\left(b-c\right)\left(c-a\right)\left(a-b\right)$
(d) none of these

Q.

If the first and the n^th term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P² = (ab)ⁿ.

Q. If $x,y,z$ are positive real numbers and $a,b,c$ are rational numbers, then the value of $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}$ is:

1. $1$
2. $-1$
3. $0$
4. None of these

Q.

The sum of first three terms of a G.P. is and their product is 1. Find the common ratio and the terms.

Q.

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Q.

The ratio of the sums of m and n terms of an A.P. is m²: n². Show that the ratio of m^th and n^th term is (2m – 1): (2n – 1).

Q. An infinite GP can have a finite sum if the common ratio is more than 1.

1. False
2. True

Q. If $5+55+555+\cdots \text{to}n\text{terms}=\frac{5}{81}({10}^{n+1}+x+yn)$, then the value of $y-x$ is

1. $1$
2. $-19$
3. $-1$
4. $19$

Q. If $a+b+c=3$ and $a>0,b>0,c>0$the greatest value of $a^2{b}^3{c}^2$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-4x+\lambda =0$ and $\gamma ,\delta$ are the roots of the equation $x^2-64x+\mu =0$ and $\alpha ,\beta ,\gamma ,\delta$ forms an increasing G.P., then the value of $\frac{\mu }{\lambda }$ is

Q. Let f : R → R and g : R → R be defined by f(x) = x + 1 and g(x) = x − 1. Show that fog = gof = I_R.

Q.

The ratio of the sums of m and n terms of an A.P. is m²: n². Show that the ratio of m^th and n^th term is (2m – 1): (2n – 1).

Q. If $x=a+\frac{a}{r}+\frac{a}{r^2}+....\infty ,y=b-\frac{b}{r}+\frac{b}{r^2}-....\infty ,andz=c+\frac{c}{r^2}+\frac{c}{r^4}+....\infty ,\left(r\right|>1,c\ne 0then\frac{xy}{z}=$
.

1. $\frac{ab}{c}$
2. $\frac{ac}{b}$
3. abc
4. $\frac{c}{ab}$

Q. ntIf x{(1+y)} + y{(1+x)} = 0, then show that, n ntn ntdy/dx= 1/(1+x)n ntn ntn nt(SN pg 315)n

Q.

If the first and the n^th term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P² = (ab)ⁿ.

Q. If a ball is dropped from a height of $40\text{m}$ and bounce back upto $60\%$ of the orginal height, then total distance travelled (in $\text{m}$) by the ball before coming to rest is

Q. If each term of an infinite G.P. is thrice the sum of the terms following it, then the common ratio of the G.P. is .

1. $\frac{1}{4}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{2}$

Q. An infinite GP can have a finite sum if the common ratio is more than 1.

1. False
2. True

Q. If $x=a+\frac{a}{r}+\frac{a}{r^2}+\cdots \infty ,$ $y=b-\frac{b}{r}+\frac{b}{r^2}-\cdots \infty$ and $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\cdots \infty$ for $|r|>1,$ then the value of $\frac{xy}{z}$ is

1. $\frac{ab}{c}$
2. $\frac{a{b}^2}{c}$
3. $\frac{ac}{b}$
4. $\frac{a^{2}b}{c}$