Sequences & Series
Sequence and Series
star_batch_jee_advanced_2025
Grade 11
Question:
Let $\{a_n\}$ consists of positive numbers and for any positive integer $n$, $\frac{a_n+2}{2} = \sqrt{2s_n}$, where $s_n = \sum_{i=1}^{n} a_i$. Then:
$a_{21} = 82$
$a_{12} = 48$
$a_{13} = 50$
$a_{14} = 54$
Step-by-Step Solution
Key Concept: Subtract consecutive recurrence relations to isolate the common difference of the arithmetic sequence.
From $\frac{a_1 + 2}{2} = \sqrt{2a_1}$, squaring gives $(a_1 - 2)^2 = 0$, so $a_1 = 2$. Using the recurrence relation $(a_n + 2)^2 = 8a_n$ and $(a_{n-1} + 2)^2 = 8a_{n-1}$, we derive $(a_n + 2)^2 - (a_{n+1} + 2)^2 = 8a_n - 8a_{n+1}$, which simplifies to $a_n - a_{n-1} = 4$. This shows the sequence is arithmetic with common difference 4.
Correct Answer: 3,4