A $G . P .$ consists of an even number of terms. The sum of all the terms is three times that of the odd terms. Determine the common ratio of the $G . P .$ .

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Solution

By given condition, we have
$S_{2 n} = 3 [T_{1} + T_{3} + T_{5} + . . . . . + T_{2 n - 1}]$
$= 3 [a + {a r}^{2} + {a r}^{4} + . . . . . . n t e r m s]$
$∴ \frac{a}{1 - r} (1 - r^{2 n}) = 3. \frac{a}{1 - r^{2}} [{1 - (r^{2})}^{n}]$
or $1 = \frac{3}{1 + r} ∴ r = 2.$

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A G.P. consists of an even number of terms. The sum of all the terms is three times that of the odd terms. Determine the common ratio of the G.P..

By given condition, we have S2n=3[T1+T3+T5+.....+T2n−1]=3[a+ar2+ar4+......n terms]∴a1−r(1−r2n)=3.a1−r2[1−(r2)n]or 1=31+r∴r=2.

A $G . P .$ consists of an even number of terms. The sum of all the terms is three times that of the odd terms. Determine the common ratio of the $G . P .$ .

By given condition, we have
$S_{2 n} = 3 [T_{1} + T_{3} + T_{5} + . . . . . + T_{2 n - 1}]$
$= 3 [a + {a r}^{2} + {a r}^{4} + . . . . . . n t e r m s]$
$∴ \frac{a}{1 - r} (1 - r^{2 n}) = 3. \frac{a}{1 - r^{2}} [{1 - (r^{2})}^{n}]$
or $1 = \frac{3}{1 + r} ∴ r = 2.$