If Σₖ₌₁¹⁰ k/(k⁴+k²+1)=m/n, where m and n are co-prime, then m+n is equal to _____________.

Solution

Maths sir · 4 steps

Factor k⁴+k²+1 into two quadratics that differ by 2k, and the sum collapses to two terms.

1. Factor the denominator. Add and subtract k²: k⁴+k²+1 = (k²+1)² - k² = (k²+k+1)(k²-k+1).
2. Split the fraction. The two factors differ by (k²+k+1)-(k²-k+1) = 2k. So the numerator k is half of that difference, and the term splits into two fractions.

   k/(k⁴+k²+1) = 1/2(1/(k²-k+1)-1/(k²+k+1))

3. Watch it telescope. Note that k²+k+1 = (k+1)²-(k+1)+1, so each negative term cancels the next positive term. The denominators run 1, 3, 7, 13, …, 111, and only the first and last survive: the sum is 1/2(1-1/111) = 1/2 × 110/111 = 55/111.
4. So. 111 = 3 × 37 shares no factor with 55 = 5 × 11, so m = 55, n = 111 and m+n = 166.

The common slip: 221. Forgetting the factor 1/2 in front gives 110/111 and m+n = 221, not 166.

The idea to keep. When you see k⁴+k²+1, write it as (k²+k+1)(k²-k+1). The two factors differ by 2k, so a numerator of k always splits into a telescoping difference.

Answer from the NTA final answer key. Solution by PrepWiser.