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Arithmetic Progressions
CH05 Question Bank
CBSE_CH05_QUESTION_BANK
Grade 10

Question:

If the sum of the first $6$ terms of an AP is $42$, and the ratio of its $10$th term to its $30$th term is $1:3$, find the first term and the common difference.

Step-by-Step Solution

Key Concept: Translate both conditions into equations in a and d, then solve simultaneously.
$S_6=42\Rightarrow\dfrac62[2a+5d]=42\Rightarrow2a+5d=14$ — (1). [1.0 Mark]

$\dfrac{a_{10}}{a_{30}}=\dfrac13\Rightarrow3(a+9d)=a+29d\Rightarrow3a+27d=a+29d\Rightarrow2a=2d\Rightarrow a=d$ — (2). [1.0 Mark]

Substituting (2) into (1): $2d+5d=14\Rightarrow7d=14\Rightarrow d=2$, so $a=2$. [1.0 Mark]

Correct Answer:
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