Question

Find sum to n terms :

3¹+3²+3³......

Answer 1

Let us assume sum of this is equal to S.

Then,

S = 3 + 33 + 333 + ....
S = 3 (1 + 11 + 111 + ....)
S = 3 [(1) + (1+10) + (1+10+100) + ... + (1+10+100+...+10^(n−1))
S = 3 [(10¹−1)/(10−1) + (10²−1)/(10−1) + (10³−1)/(10−1) + ... + (10ⁿ−1)/(10−1)]
S = 3/9 [(10¹ + 10² + 10³ + ... + 10ⁿ) − n]
S = 1/3 [10 (10ⁿ−1)/(10−1) − n]
S = 1/3 [10 (10ⁿ−1)/9 − 9n/9]
S = 1/27 (10 (10ⁿ−1) − 9n)