Question

If the and 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. $a ^ { 2 } = b c$
• C. $b ^ { 2 } = a c$
• D. $c ^ { 2 } = a b$

Answer 1

Answer: (C)

$b ^ { 2 } = a c$

Answer 2

Let first term of G.P. $= A$ and common ratio $= r$

We know that $n ^ { t h }$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 $b ^ { 2 } = \left( A r ^ { 6 } \right) ^ { 2 } = A ^ { 2 } r ^ { 12 }$ and $a c = \left( A r ^ { 3 } \right) \left( A r ^ { 9 } \right) = A ^ { 2 } r ^ { 12 }$

Aliter : As we know, if $p , q , r$ in A.P., then $p ^ { t h } , q ^ { t h } , r ^ { t h }$ terms of a G.P. are always in G.P., therefore, $a , b , c$ will be in G.P. $b ^ { 2 } = a c$.