Tag: geometric sequences

Questions Related to geometric sequences

The sum of $6^{th}$ term in the geometric series $4, 12, 36...$ is

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $1456$

  2. $2456$

  3. $3456$

  4. $4456$

Show answer
Correct Option: A
Explanation:

Given sequence is $4,12, 36$
To find the sum of the first $S _n$ terms of a geometric sequence using the formula
Here $a = 4, r = 3, n = 6$
We know $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$\Rightarrow S _6 = \dfrac{4(1-3^{6})}{1-3}$
$\Rightarrow S _6 = \dfrac{-2912}{-2}$
$\Rightarrow S _6 = 1456$

What is the sum of G.P. $1, 3, 9, 27,.....$ up to $7$ numbers?

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $1093$

  2. $2093$

  3. $3093$

  4. $4093$

Show answer
Correct Option: A
Explanation:

Given, $a = 1, r = 3$
$S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _7 = \dfrac{1(1-3^7)}{1-3}$
$ = \dfrac{1(1-3^7)}{-2}$
$ = 1093$

What is the sum of the first five terms of the geometric sequence $5, 15, 45, ... $?

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $105$

  2. $305$

  3. $505$

  4. $605$

Show answer
Correct Option: D
Explanation:

Given series is $5,15,45,...$
Here first term $a = 5$ and common ratio $=r = 3$
$S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _{5} = \dfrac{5(1-(3)^{5})}{1-(3)}$
$ = \dfrac{5(1-(3)^{5})}{-2}$
$ = 605$

Find $4 + 12 + 36 +..... $ upto $6$ terms.

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $164$

  2. $264$

  3. $364$

  4. $464$

Show answer
Correct Option: C
Explanation:

Given series is $4+12+36+....$ upto $6$ terms
Here $a = 4, r = 3$
We know $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _{6} = \dfrac{1(1-(3)^{6})}{1-(3)}$
$ = \dfrac{1(1-(3)^{6})}{-2}$
$ = 364$

If the first term and common ratio is $5$, find the sum of 8th term of infinite G.P.

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $188280$

  2. $288280$

  3. $388280$

  4. $488280$

Show answer
Correct Option: D
Explanation:

Given, first term $=a = 5$, common ratio $=r = 5$, $n=8$
We know $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _{8} = \dfrac{5(1-(5)^{8})}{1-(5)}$
$ = \dfrac{5(1-(5)^{8})}{-4}$
$ = 488280$

The value of $1 + 2 + 4 + 8....$ of G.P., where $n=6$  is

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $61$

  2. $62$

  3. $63$

  4. $64$

Show answer
Correct Option: C
Explanation:
Given series is $1+2+4+8+....$
Here $a = 1, r = 2$
Also given $n=6$
We know $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _{6} = \dfrac{1(1-(2)^{6})}{1-(2)}$
$ = \dfrac{1(1-(2)^{6})}{-1}$
$ = 63$

Calculate sum of eleventh term of the geometric sequence $3, 6, 12, 24, ... $

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $3141$

  2. $6141$

  3. $2141$

  4. $5141$

Show answer
Correct Option: B
Explanation:

Given series is $3,6,12,24,....$
Here first term $=a = 3$ and common ratio $=r = 2$
$S _n = \dfrac{a _1(1-r^n)}{1-r}$
$S _{11} = \dfrac{3(1-(2)^{11})}{1-(2)}$
$ = \dfrac{3(1-(2)^{11})}{-1}$
$ = 6141$

The sum of first $n$ terms of an G.P. is

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $S _n = \cfrac{a _1(1-r^n)}{1-r}$

  2. $S _n = \cfrac{a _1(1+r^n)}{1-r}$

  3. $S _n = \cfrac{a _1(1-r^n)}{1+r}$

  4. $S _n = \cfrac{a _1(1-r^n)}{r-1}$

Show answer
Correct Option: A
Explanation:

A GP can be written as:

$a,ar,ar^2, ar^3..............,ar^{n-1}$
$\text{sum} = a+ar+ar^2+ar^3+.........+ar^{n-1}$
$\text{sum} = a(r^{n-1}+r^{n-2}+r^{n-3}+r^{n-4}+...........+r+1)$
We know that:
$\dfrac{x^n -1}{x-1} = x^{n-1}+x^{n-2}+x^{n-3}+.............+x+1$
Thus $\text{sum} = a\left (\dfrac{r^n -1}{r-1}\right)$

A rubber ball is dropped from a height of $10$ meters. If the ball always rebounds $\dfrac {4}{5}$ the distance it has fallen, calculate, how far, in meters, will the ball have travelled at the moment it hits the ground for the fourth time?

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $4.10$

  2. $5.12$

  3. $29.52$

  4. $43.92$

  5. $49.04$

Show answer
Correct Option: E
Explanation:
  • for the first time it hits the ground , it travels $10$ m
  • for the second time it hits the ground , it travels $2 \times 4/5 \times 10 = 16+10 =26$
  • for the third time it hits the ground , it travels $ 2\times 4/5 \times 4/5 \times 10 +26 =64/5+26 = 38.8$
  • for the fourth time it hits the ground , it travels $2 \times 4/5 \times 4/5 \times 4/5 \times 10 +38.8 = 256/25 + 38.8 =49.04$

Find the sum of odd integers between $1$ and $1000$ which are divisible by $3$.

maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

  1. $83667$

  2. $54954$

  3. $99994$

  4. $79894$

Show answer
Correct Option: A
Explanation:
Odd integers divisible by 3 are: $3,9,15,21.....999$
Let $a=3$   and   $d=6$
$\therefore T _n=999=3+(n-1)6$
$\Rightarrow n-1=\dfrac{996}{6}=166$
$\therefore n=167$
$\therefore S _n=\dfrac{167}{2}[2\times 3+(167-1)6]=83667$