Question

By induction:

1. Prove that Σⁿ_i=1(2i − 1) = n²

2. Prove thatΣⁿ_i=1 i² = n(n+1)(2n+1) / 6 .

Answer 1

i think u want this proof. your question 1 is not clear. Also 2nd is not clear

but i tried my best
1 ² + 2 ² + 3 ² + ... + n ² = n (n + 1) (2n + 1)/ 26
STEP 1: We first show that p (1) is true.
Left Side = 1 ² = 1
Right Side = 1 (1 + 1) (2*1 + 1)/ 6 = 1
Both sides of the statement are equal hence p (1) is true.
STEP 2: We now assume that p (k) is true
1 ² + 2 ² + 3 ² + ... + k ² = k (k + 1) (2k + 1)/ 6
and show that p (k + 1) is true by adding (k + 1) ² to both sides of the above statement
1 ² + 2 ² + 3 ² + ... + k ² + (k + 1) ² = k (k + 1) (2k + 1)/ 6 + (k + 1) ²
Set common denominator and factor k + 1 on the right side
= (k + 1) [ k (2k + 1)+ 6 (k + 1) ] /6
Expand k (2k + 1)+ 6 (k + 1)
= (k + 1) [ 2k ² + 7k + 6 ] /6
Now factor 2k ² + 7k + 6.
= (k + 1) [ (k + 2) (2k + 3) ] /6
We have started from the statement P(k) and have shown that
1 ² + 2 ² + 3 ² + ... + k ² + (k + 1) ² = (k + 1) [ (k + 2) (2k + 3) ] /6
Which is the statement P(k + 1).