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

Evaluate the sum of the first nine terms of the geometric sequence $5, 10, 20,...$

  1. $1555$
  2. $2555$
  3. $3555$
  4. $4555$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given sequence is $5,10,20,....$
To find the sum of the first $S _n$ terms of a geometric sequence using the formula
Here $a = 5, r = 2, n = 9$
We know $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$\Rightarrow S _9 = \dfrac{5(1-2^{9})}{1-2}$
$\Rightarrow S _9 = \dfrac{-2555}{-1}$
$\Rightarrow S _9 = 2555$

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

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

  1. $1456$
  2. $2456$
  3. $3456$
  4. $4456$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $S _n = \dfrac{a _1(1-r^n)}{1-r}$
$\Rightarrow S _6 = \dfrac{4(1-3^{6})}{1-3}$
$\Rightarrow S _6 = \dfrac{-2912}{-2}$
$\Rightarrow S _6 = 1456$

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

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

  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. $S _n = \cfrac{a _1(1-r^n)}{r-1}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $\text{sum} = a\left (\dfrac{r^n -1}{r-1}\right)$