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The 8th term of a GP is -7/32. Find its common ratio if its first term is 28.

June 15, 2026Mathematics

ScanSolve AI Answer

Answer

-\frac{1}{2}

4 steps

The formula for the $n$ -th term of a Geometric Progression (GP) is $T_n = ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

Identify the given values.
The 8th term ( $T_8$ ) is $-\frac{7}{32}$ .
The first term ( $a$ ) is 28.
The number of terms ( $n$ ) is 8.
Substitute the given values into the formula for the 8th term.
$T_8 = ar^{8-1}$ $-\frac{7}{32} = 28r^7$
Solve for the common ratio $r$ .
Divide both sides by 28:
$r^7 = \frac{-\frac{7}{32}}{28}$ $r^7 = -\frac{7}{32 \times 28}$
Simplify the fraction:
$r^7 = -\frac{1}{32 \times 4}$ $r^7 = -\frac{1}{128}$
Find the 7th root of $-\frac{1}{128}$ .
Since $2^7 = 128$ , we have:
$r^7 = -\left(\frac{1}{2}\right)^7$ $r = -\frac{1}{2}$
The common ratio is $-\frac{1}{2}$ .