I need someone's idea here... im trying to prove a pattern we formulated with my classmate as we deal with the even powers of natural numbers that can be express as a sum of consecutive odd numbers starting from 1 here's the formula we come up with$$\sum_{i=1}^{n^{k}} (2i - 1) = n^{2k}$$ For k = 1, $$\sum_{i=1}^{n} (2i - 1) = n^{2}$$ Which can be easily proven by mathematical induction but when k = 2, it will become,$$\sum_{i=1}^{n^{2}} (2i - 1) = n^{4}$$ and when im doing the mathematical induction i got stuck for some terms in between n and n + 1 such as $1 + 3 + 5 + ...+ (2n^{k} - 1)$ + (some in between terms i cant express in summation) + $(2(n +1)^{k} -1) = (n +1)^{4}$
Actually i want to prove by mathematical induction the general k, but i struggle so much that i decided to manually assume values for k = 1,2,3... but it seems that i cant manage to deal with k = 2 . Any ideas? Thanks in advance!
Your statement can easily be shown without induction, simply using the fact that
$$\sum_{i=1}^N = \frac{N(N+1)}{2}$$
and $$\sum_{i=1}^N 1 = N.$$
This is because
$$\sum_{i=1}^{n^k}(2i-1) = 2\cdot\sum_{i=1}^{n^k} i - \sum_{i=1}^{n^k} 1.$$