Question

4. The product of three terms in G.P is 1000, what is its middle term?
(1) 12
(i) 14
(iii) 16​

Answer 1

$\large{\underline{\underline{\red{\sf{Given:}}}}}$

• $\tt{Product\:of\:3\:terms\:of\:GP\:is\:1000}$

$\large{\underline{\underline{\red{\sf{To\:Find:}}}}}$

• $\tt{The\:middle\:term\:of\:GP}$

$\large{\underline{\underline{\red{\sf{Answer:}}}}}$

$\purple{\rm{Geometric\: Progression}}$

$\sf{Numbers\:are\:said\:to\:be\:in\:GP\:if\:when }$$\sf{\:ratio\:of\:any\: quantity\:of\: number\: that }$$\sf{follows \:it\:is\: same.}$

$\tt{Given\:that\: product\:of\:3\:terms\:of\:a\:GP\:is\:1000.}$

$\tt{Let\:the\:terms\:be\:\green{\dfrac{a}{r},\:a,\:ar.}}$

$\bf{Where}$

• $\tt{r\:is\: common\:ratio.}$
• $\tt{\dfrac{a}{r}\:is\:first\:term.}$
• $\tt{{a}\:is\: middle\:term.}$
• $\tt{{ar}\:is\:third\:term.}$

$\purple{\rm{As\:per\:given\:in\:question:}}$

$\tt{\implies (\dfrac{a}{\cancel{r}})\times (a)\times (a\cancel{r})=1000 .}$

$\tt{\implies a^3 \:\:=\:\:1000 .}$

$\tt{\implies a=\sqrt[3]{1000}}$

$\underline{\boxed{\red{\sf{\longmapsto a=10}}}}$

$\tt{Earlier\:we\:assumed\:middle\:term\:to\:be\:a.}$

$\tt{And\:we\:got\:value\:of\:a\:as\:10.}$

$\pink{\tt{\underset{\purple{Middle\:term}}{\underbrace{Required\:answer\:is\:10.}}}}$

_____________________________________