Find the sum of the n terms of the series 1+3+7+15+31+⋯n terms.
2(n+1)−2−n
We see that the differences between successive a term is 2,4,8,16,⋯
This is a GP.
Sn=1+3+7+15+31+⋯Tn−1+Tn
⋯(1)
Sn=1+3+7+15+⋯Tn−1+Tn
⋯(2)
[Now, we write the Sn in such a way that 1st term of equation 2 comes under 2nd term of equation 1]
Then, Subtracting both, we get,
0=1+2+4+8+16+⋯+(Tn+Tn−1)−Tn
Tn=1+2+4+8+16+⋯n terms
we get, Tn = 1.(2n−1)2−1
Hence sum of the n terms
Sn=∑tn=∑ni=1(2i−1)
Sn=∑2n−∑1=2(n+1)−2−n
Aliter:
§n=1+3+7+15+⋯+nthterm=(21−1+22−1+23−1+⋯+2n−1=21+22+23+⋯+2n−n=2×1−2n1−2−n=−2+2n+1−n