The first four terms of sequence {an} are 3 / 2,9 / 4,25 / 8,65 / 16. Try to find the general term formula and the sum of the first n terms

The first four terms of sequence {an} are 3 / 2,9 / 4,25 / 8,65 / 16. Try to find the general term formula and the sum of the first n terms

Look at the denominator first
2^1,2^2,2^3...2^n
Look at molecules again
3=2*1+1
9=4*2+1
25=8*3+1
65=16*4+1
in other words
Numerator = denominator * n + 1 = n * 2 ^ n + 1
The general term formula is not unique
Is an = (n * 2 ^ n + 1) \ (2 ^ n)
As for summation, we can simplify the general term first
an=n+(1/2)^n
The former part n can be calculated by equal difference
The latter part (1 / 2) ^ n can be calculated proportionally