Question

What is the sum of the first 10 terms given 1, 1/3, 1/9,1/27..?

Answer 1

`\color{red}{3/2(1-(1/3)^{10})\approx1.499974597`

Explanation:

Given series `1, 1/3, 1/9, 1/27, \ldots` is a G.P. with the first term `a=1` & common ratio `r=\frac{1/3}{1}=\frac{1/9}{1/3}=\frac{1/27}{1/9}=\ldots =1/3`

Now, the sum of given series up to `n=10` terms

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

`=\frac{1((1/3)^{10}-1)}{1/3-1}`

`=\frac{(1/3)^{10}-1}{-2/3}`

`=3/2(1-(1/3)^{10})`

`approx1.499974597`

Answer 2

`S_10=3/2((3^10-1)/3^10)~~1.4999`

Explanation:

Here,

`1,1/3,1/9,1/27,...,`

The first term `=a_1=1`

The common ratio `=r=(1/3)/1=(1/9)/(1/3)=(1/27)/(1/9)=1/3`

So, the given sequence is Geometric sequence with

`color(blue)(a_1=1 and r=1/3`

The sum of first `n-terms` of geometric sequence is :

`color(blue)(S_n=(a_1(1-r^n) )/(1-r) )and` we have `color(blue)(n=10`

So,

`S_10=(1(1-(1/3)^10))/(1-1/3)`

`S_10=((1-1/3^10))/(2/3)`

`S_10=3/2((3^10-1)/3^10)`

`S_10=3/2(59048/59049)`

`S_10~~1.4999`