Let (P(n): 1^2+2^2+3^2+ +n^2 ) (= 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 ), for all (n N ). Then which of the following is correct?

Step-by-step solution

Given statement ( aligned & P(n)=1^2+2^2+3^2+ +n^2= & 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 aligned ) ( ) We know that, ( aligned & 1^2+2^2+3^2+ +n^2= n(n+1)(2 n+1) 6 & 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 & =(n-1)(n-2) (n-2020)+ n(n+1)(2 n+1) 6 aligned ) will be ( n(n+1)(2 n+1) 6 ) if (n=1,2,3, ., 2020 ) only. Therefore, (P(n) ) is true for all (n 2020 ). Hence, option (c) is correct.