Question

find the sum of the 20 terms of the geometric series 5/2 +5/6+5/18+.....

Answer 1

To find the sum of first 20 terms of the geometric series first we have to find the value of r. Then only we can decide which formula should be used in this problem.

 r = t₂/t₁                 a = 5/2   and  n = 20 

 r = (5/6)/(5/2)

   = (5/6) x (2/5)

 r = 1/3  here r < 1 

sn = a(1- rn)⁄(1 - r)

So, S₂₀ = (5/2) [1- (1/3)^₂₀]/[1-(1/3)]

          = (5/2) [1- (1/3)^₂₀]/[2/3]

          = (5/2) x  (3/2) [1- (1/3)^₂₀]

          = (15/4) [1 - (1/3)^₂₀]

Answer 2

Answer:

The sum of geometric series is 15/4 (1-$1/3^{20}$) .

Step-by-step explanation:

As per the data given in the question,

we have to calculate that the sum of the 20 terms of the geometric series

As per the questions

It is given that geometric series 5/2 +5/6+5/18+.....

We have to find the common ratio r first.

In the given geometric series, the first term is $a_{1}$ = 5/2 and second term is $a_{2}$ = 5/6 and so on.

We find the common ratio r by dividing the second term by first term

r= $\frac{5/6}{5/2}=\frac{5}{6}$×$\frac{2}{5}$=$\frac{1}{3}$

We know that the sum of an geometric series with first term a and common ratio r is

$s_{n}=\frac{a(1-r^{n})}{1-r}$

similarly $s_{20}=\frac{5/2 (1-(1/3^{20}) }{1-1/3}$=    $\frac{5/2 (1-1/3^{20}) }{2/3}$ =5/2 ×3/2 (1-$1/3^{20}$) = 15/4 (1-$1/3^{20}$)

Hence, the sum of geometric series is 15/4 (1-$1/3^{20}$) .

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