Problem: The sum of the first n terms of the sequence 1,(1+2),(1+2+22),…(1+2+22+…+2n−1) in terms of n is
Answer Choices:
A. 2n
B. 2n−n
C. 2n+1−n
D. 2n+1−n−2
E. n⋅2n
Solution:
The kth term of the given sequence is equal to 2k−1 so that the sum of the first n terms may be written as
(21−1)+(22−1)+(23−1)+⋯+(2n−1)
=(2+22+23+⋯+2n)−(1+1+1+⋯+ to n terms )
=(2n+1−2)−n=2n+1−n−2.