Sequences & Series
AP — product of middle terms
nta_pyq_2023_jan
Grade 11
Question:
Let $a_1 = 8$, $a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is ______.
Step-by-Step Solution
Key Concept: From sum of first four terms: $4a_1 + 6d = 50$, so $d = \frac{50-32}{6} = 3$. Use sum of last four terms to find $n$, then identify middle terms.
$d = 3$. Last four terms sum: $4a_n - 6d = 170 \Rightarrow a_n = \frac{188}{4} = 47$. $a_n = 8 + (n-1)3 = 47 \Rightarrow n = 14$. Middle terms: $a_7 = 8+18 = 26$ and $a_8 = 29$. Product $= 26 \times 29 = 754$.
Correct Answer: 754