Sequences & Series
Recurrence Relations
Grade 11
Question:
<p>First term of a sequence is 1 and the <span>\((n+1)\)</span>-th term is obtained by adding <span>\((n+1)\)</span> to the <span>\(n\)</span>-th term for all natural numbers <span>\(n\)</span>. The 6th term of the sequence is</p>
<p>(a) 7</p>
<p>(b) 13</p>
<p>(c) 21</p>
<p>(d) 27</p>
Step-by-Step Solution
Key Concept: Use the recurrence relation to iteratively compute terms by repeatedly adding successive natural numbers.
<p><strong>Step 1:</strong> Given <span>$a_1 = 1$</span> and <span>$a_{n+1} = a_n + (n+1)$</span></p><p><strong>Step 2:</strong> Calculate successive terms:</p><ul><li><span>$a_2 = a_1 + 2 = 1 + 2 = 3$</span></li><li><span>$a_3 = a_2 + 3 = 3 + 3 = 6$</span></li><li><span>$a_4 = a_3 + 4 = 6 + 4 = 10$</span></li><li><span>$a_5 = a_4 + 5 = 10 + 5 = 15$</span></li><li><span>$a_6 = a_5 + 6 = 15 + 6 = 21$</span></li></ul><p><strong>∴ Answer is (c) 21</strong></p>
Correct Answer: C