Let `S_(n)` denote the sum of n terms of the series
`1^(2)+3xx2^(2)+3^(2)+3xx4^(2)+5^(2)+3xx6^(2)+7^(2)+ . . . .`
Statement -1: If n is odd, then `S_(n)=(n(n+1)(4n-1))/(6)`
Statement -2: If n is even, then `S_(n)=(n(n+1)(4n+5))/(6)`

Options

A. Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.
B. Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.
C. Statement -1 is true, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Answer

A

Explanation

Let n be even and let n=2m. Then,
`S_(n)=1^(2)+3xx2^(2)+3^(2)+3xx4^(2)+5^(2)+3xx6^(2)+ . . . +(2m-1)^(2)+3xx(2m)^(2)`
`rArr" "S_(n)={1^(2)+3^(2)+5^(2)+ . . . .+(2m-1)^(2)}+3{2^(2)+4^(2)+6^(2)+ . . . +(2m)^(2)}`
`rArr" "s_(n)=underset(r=1)overset(2m)sumr^(2)+2underset(r=1)overset(m)sum(2r)^(2)`
`rArr" "S_(n)=(2m(2m+1)(4m+1))/(6)+(8m(m+1)(2m+1))/(6)`
`rArr" "S_(n)(2m(2m+1))/(6){(4m+1)+4(m+1)}`
`rArr" "S_(n)=(1)/(6)(2m)(2m+1)(4(2m)+5)=(n(n+1)(4n+5))/(6)`
So, statement -2 is true. If n is odd, then
`S_(n)=1^(2)+3xx2^(2)+3^(2)+3xx4^(2)+ . . . +3xx(n-1)^(2)+n^(2)`
`rArr" "S_(n)=((n-1)(n-1+1)(4(n-1)+5))/(6)+n^(2)`
Replacing n by n-1 in statement-2.
`rArr" "S_(n)(n(n-1)(4n+1))/(6)+n^(2)`
`rArr" "S_(n)(n)/(6){(n-1)(4n+1)+6n}`
`rArr" "S_(n)=(n)/(6)(4n^(2)+3n-1)=(n(n+1)(4n-1))/(6)`
So, statement -1 is true and statement -2 is a correct explanation for statement -1.