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8. Sequences and Series
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If the ${4^{th}},\;{7^{th}}$ and ${10^{th}}$ terms of a $G.P.$ be $a,\;b,\;c$ respectively, then the relation between $a,\;b,\;c$ is
A
$b = \frac{{a + c}}{2}$
B
${a^2} = bc$
C
${b^2} = ac$
D
${c^2} = ab$
Solution
(c) Let first term of $G.P.$ $ = A$ and common ratio $ = r$
We know that ${n^{th}}$ term of $G.P. =$ $A{r^{n – 1}}$
Now ${t_4} = a = A{r^3},\;{t_7} = b = A{r^6}$and ${t_{10}} = c = A{r^9}$
Relation ${b^2} = ac$ is true because ${b^2} = {(A{r^6})^2} = {A^2}{r^{12}}$
and $ac = (A{r^3})(A{r^9}) = {A^2}{r^{12}}$
Aliter : As we know, if $p,q,r$ in $A.P.$,
then $p^{th},q^{th},r^{th}$ terms of a $G.P.$ are always in $G.P.$,
therefore, will be in $G.P.$
Standard 11
Mathematics
