The sum of n terms of the series +1+ 4 5 + 7 5^2 + 10 5^3 + is

Step-by-step solution

Clearly, the given series is an arithmetico-geometric series whose corresponding AP and GP. are respectively, 1,4,7,10, and 1, 1 5 , 1 5^2 , 1 5^3 , The n^ th term of A P=[1+(n-1) 3]=3 n-2 The n^ th term of GP = [1 ( 1 5 )^ n-1 ]= ( 1 5 )^ n-1 So, the n^ th term of the given series is (3 n-2) 1 5^ n-1 = 3 n-2 5^ n-1 Let, aligned S_n=1+ 4 5 + 7 5^2 & + 10 5^3 + & + 3 n-5 5^ n-2 + 3 n-2 5^ n-1 aligned ...(i) 1 5 S_n= 1 5 + 4 5^2 + 7 5^3 + + (3 n-5) 5^ n-1 + 3 n-2 5^n ....(ii) Subtracting (ii) from (i), we get aligned