Question

Find the sum of the series upto n terms.
(1+x)+(1+x+x2)+(1+x+x2+x3)+....​

Answer 1

Answer:

The sum of the given series will be $n(1+x)+ (n-1)x^{2} +(n-2)x^{3}..+2x^{n-1} +x^{n}$

Step-by-step explanation:

We are given to find the sum of n terms where the terms are increasing this way :-

(1+x)+(1+x+x2)+(1+x+x2+x3)+

Observe carefully and 1 and x will be in all the terms which will come next

This means that there will be a total of n 1's and n x's now the next term is $x^{2}$ it is coming from the 2nd term and it will be there till the nth terms which means it will have a total of (n-1) terms similarly $x^{3}$ will come (n-2) times and so on

Now it will end it with 2 $x^{n-1}$ and 1 $x^{n}$ because $x^{n-1}$ will only be there in last term and the term preceeding it and  $x^{n}$ will only be present in the last term

Therefore the sum becomes $n(1+x)+ (n-1)x^{2} +(n-2)x^{3}..+2x^{n-1} +x^{n}$