Sn = n^2 + 5n + 94 and K = S6 – S5 + S4 – S3 + S2 – S1. What is the

MANHATTAN GMAT OFFICIAL SOLUTION:

The expression for K becomes more manageable if we insert parentheses:
K = (S6 – S5) + (S4 – S3 )+ (S2 – S1)

K involves two expressions of the form \((S_{n + 1} - S_n)\), so a general rule for \((S_{n + 1} - S_n )\) might be helpful. We know that \(S_n = n^2 + 5n + 94\), so

\(S_{n + 1} = (n + 1)^2 + 5(n + 1) + 94\)
\(S_{n + 1} = (n^2 + 2n + 1) + (5n + 5) + 94\)
\(S_{n + 1} = n^2 + 7n + 100\)

In general, \((S_{n + 1} - S_n) = (n^2 + 7n + 100) - (n^2 + 5n + 94) = 2n + 6\).

Applying this rule to the grouped terms in K:
K = (S6 – S5) + (S4 – S3) + (S2 – S1)
= (2 × 5 + 6) + (2 × 3 + 6) + (2 × 1 + 6)
= 16 + 12 + 8
= 36

Alternatively, we could just plug into the Sn term definition, and identify common terms as we work:
K = S6 – S5 + S4 – S3 + S2 – S1= (6^2 + 5(6) + 94) –(5^2 + 5(5) + 94) + (4^2 + 5(4) + 94) – (3^2 + 5(3) + 94) + (2^2 + 5(2) + 94) – (1^2 +5(1) + 94)= (6^2 – 5^2 + 4^2 – 3^2 + 2^2 – 1^2) + 5(6 – 5 + 4 – 3 + 2 – 1)= (36 – 25 + 16 – 9 + 4 – 1) + 5(3)= (21) + (15)= 36

The correct answer is E.