Problem: The sum of all numbers of the form 2k+12 k+12k+1, where kkk takes on integral values from 111 to nnn is:
Answer Choices:
A. n2n^{2}n2
B. n(n+1)n(n+1)n(n+1)
C. n(n+2)n(n+2)n(n+2)
D. (n+1)2(n+1)^{2}(n+1)2
E. (n+1)(n+2)(n+1)(n+2)(n+1)(n+2) Solution:
s=3+5+7+⋯+(2n+1)=n2(3+2n+1)=n(n+2)s=3+5+7+\cdots+(2 n+1)=\dfrac{n}{2}(3+2 n+1)=n(n+2)s=3+5+7+⋯+(2n+1)=2n​(3+2n+1)=n(n+2).