Question

If sumk=1ⁿ k(k+1)(k-1)=pn⁴+qn³+tn²+sn, where p,q,t and s are constants, then the value of s is equal to

• A. -(1)/(4)
• B. -(1)/(2)
• C. (1)/(2)
• D. (1)/(4)

Hint:

Reduce polynomial summands first, then apply standard summation formulas.

Answer 1

The Correct Option is B

Step 1: Simplify the general term: k(k+1)(k-1)=k³-k Step 2: Use standard summation formulas: sumk=1ⁿ k³=((n(n+1))/(2))², sumk=1ⁿ k=(n(n+1))/(2) Step 3: Hence, sumk=1ⁿ (k³-k) =(n⁴)/(4)+(n³)/(2)-(n²)/(4)-(n)/(2) Step 4: Comparing with pn⁴+qn³+tn²+sn, we get: s=-(1)/(2)