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Worked solution

Recover the first term and difference from two terms

The third term of an arithmetic progression is 10 and the tenth term is 24. Find the first term and the common difference.

CJC · Sequences and Series · Example 8

Answer

a=6a=6, d=2d=2.

Compare your reasoning

Write both given terms

a+2d=10(1),a+9d=24(2).\begin{aligned}a+2d&=10\qquad\text{(1)},\\a+9d&=24\qquad\text{(2)}.\end{aligned}

The source labels these equations(1) and (2).

Subtract the equations

(2)−(1):7d=14⟹d=2\text{(2)}-\text{(1)}:\quad7d=14\quad\Longrightarrow\quad d=2

This eliminates the first term.

Substitute back into equation one

a=10−2(2)=10−4=6a=10-2(2)=10-4=6

Retain the source substitution.

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