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An arithmetic progression has its 12th term equal to 7 and its 15th term equal to 9. Find the first term and the common difference.

May 14, 2026Mathematics

ScanSolve AI Answer

Answer

-\frac{1}{3}

\frac{2}{3}

3 steps

Write down the given information using the formula for the $n^{th}$ term of an A.P., $T_n = a + (n-1)d$ .
The twelfth term is 7:
$T_{12} = a + (12-1)d \implies a + 11d = 7 \quad (*)$
The fifteenth term is 9:
$T_{15} = a + (15-1)d \implies a + 14d = 9 \quad (**)$
Subtract equation $(*)$ from equation $(**)$ to find the common difference $d$ .
$(a + 14d) - (a + 11d) = 9 - 7$ $3d = 2$ $d = \frac{2}{3}$
Substitute the value of $d$ into equation $(*)$ to find the first term $a$ .
$a + 11\left(\frac{2}{3}\right) = 7$ $a + \frac{22}{3} = 7$ $a = 7 - \frac{22}{3}$ $a = \frac{21}{3} - \frac{22}{3}$ $a = -\frac{1}{3}$
The first term is $\boxed{-\frac{1}{3}}$ and the common difference is $\boxed{\frac{2}{3}}$ .

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