Question

If n is odd, then (1+3+5+7+9+...... to n terms) is equal to
(a) n²+1
(b) n²-1
(c) n³
(d) n²​

Answer 1

Step-by-step explanation:

Any odd positive number is in the form of (4p+1)or(4p+3) for some integer P.

letn=4p+3n2−1=(4p+1)2−1=16p2+8p+1−1=8p(2p+1)⇒n2−1isdivisibleby8n2−1=(4p+3)2−1=16p2+24p+9−1=16p2+24p+8=8(2p2+3p+1)⇒n2−1isdivisibleby8

Therefore, n2−1 is divisible by 8 if n is an odd positive integer.