Question

If 1.3+3.3^2+5.3^3+7.3^4...upto n terms is equal to 3+(n-1).3^b then b is equal to?

Answer 1

1.3+2.32+3.33+⋯+(n).3n=(2n−1)(3)n+1+34

The result is true for n = 1

(2n−1)(3)n+1+34=(2−1)(3)1+1+34

=(1)(3)2+34=124=3=1.3

Let the result be true for n = k. That is

1.3+2.32+3.33+⋯+(k).3k=(2k−1)(3)k+1+34

Now we need to prove that the result is also true for n = k + 1. That is  

1.3+2.32+3.33+⋯+(n).3k+(k+1).3k+1=(2k+1)(3)k+2+34

by our supposition.

1.3+2.32+3.33+⋯+(k).3k=(2k−1)(3)k+1+34

adding(k+1).3k+1on both sides.

1.3+2.32+3.33+⋯+(k).3k+(k+1).3k+1

=(2k−1)(3)k+1+34+(k+1).3k+1

=(2k−1)(3)k+1+3+4(k+1).3k+14

=(2k−1)(3)k+1+4(k+1).3k+1+34

=(3)k+1{(2k−1)+4(k+1)}+34

=(3)k+1{(6k+3)}+34

=(3)k+1{3(2k+1)}+34

=(2k+1)(3)k+2+34

∴The result is also true for n = k + 1. Hence by the  

principal of mathematical induction the result is true for alln∈Z+