Search any question & find its solution
Question:
Answered & Verified by Expert
If the $4^{\text {th }}, 7^{\text {th }}$ and $10^{\text {th }}$ terms of a G.P. be ${ }^{a, b, c}$ respectively, then the relation between $a, b, c$ is
Options:
Solution:
2944 Upvotes
Verified Answer
The correct answer is:
$b^2=a c$
Let first term of G.P. $=A$ and common ratio $=r$
We know that $n^{\text {th }}$ 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 $b^2=a c$ is true because $b^2=\left(A r^6\right)^2=A^2 r^{12}$ $a c=\left(A r^3\right)\left(A r^9\right)=A^2 r^{12}$
Aliter : As we know, if $p, q, r$ in A.P., then $p^{\text {th }}, q^{\text {th }}, r^{\text {th }}$ terms of a G.P. are always in G.P., therefore, $a, b, c$ will be in G.P. i.e. $b^2=a c$.
We know that $n^{\text {th }}$ 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 $b^2=a c$ is true because $b^2=\left(A r^6\right)^2=A^2 r^{12}$ $a c=\left(A r^3\right)\left(A r^9\right)=A^2 r^{12}$
Aliter : As we know, if $p, q, r$ in A.P., then $p^{\text {th }}, q^{\text {th }}, r^{\text {th }}$ terms of a G.P. are always in G.P., therefore, $a, b, c$ will be in G.P. i.e. $b^2=a c$.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.