JEE Main & Advanced Mathematics Geometric Progression Geometric Mean

Geometric Mean

Category : JEE Main & Advanced

If $a,G,b$ are in G.P., then G is called G.M. between $a$ and $b$.

(1) If $a,\,{{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}},\,b$ are in G.P. then ${{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},....\,{{G}_{n}}$ are called n G.M.’s between $a$ and $b$.

(2) Insertion of geometric means : (i) Single G.M. between a and b : If a and b are two real numbers then single G.M. between $a$ and $b$$=\sqrt{ab}$.

(ii) n G.M.’s between a and b : If ${{G}_{1}},\,{{G}_{2}},\,{{G}_{3}},\,......,\,{{G}_{n}}$ are n G.M.’s between a and b, then

${{G}_{1}}=ar=a{{\left( \frac{b}{a} \right)}^{\frac{1}{n+1}}}$, ${{G}_{2}}=a{{r}^{2}}=a{{\left( \frac{b}{a} \right)}^{\frac{2}{n+1}}}$,

${{G}_{3}}=a{{r}^{3}}=a{{\left( \frac{b}{a} \right)}^{\frac{3}{n+1}}}$, ……………….., ${{G}_{n}}=a{{r}^{n}}=a{{\left( \frac{b}{a} \right)}^{\frac{n}{n+1}}}$.

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