Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>If 100 times the 100th term of an A.P. with non-zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is</p>
<p>(1) -150</p>
<p>(2) 150 times its 50th term</p>
<p>(3) 150</p>
<p>(4) Zero</p>
Step-by-Step Solution
Key Concept: Use the general term formula a_n = a + (n-1)d to convert the given condition into an equation, then solve for the relationship between a and d to find a_150.
<p><strong>Step 1:</strong> Write the general term formula: a_n = a + (n-1)d, where a is the first term and d is the common difference (d ≠ 0).</p><p><strong>Step 2:</strong> Express the given terms: a_100 = a + 99d and a_50 = a + 49d</p><p><strong>Step 3:</strong> Apply the given condition 100·a_100 = 50·a_50:<br/>100(a + 99d) = 50(a + 49d)<br/>100a + 9900d = 50a + 2450d<br/>50a + 7450d = 0<br/>50a = -7450d<br/>a = -149d</p><p><strong>Step 4:</strong> Find a_150 using a = -149d:<br/>a_150 = a + 149d<br/>a_150 = -149d + 149d<br/>a_150 = 0</p><p>∴ Answer: D (The 150th term is 0)</p>
Correct Answer: D