Question

the 3rd term of Gp is 1/2. If its 4th term is 4 times its 6th term find the 5th term​

Answer 1

Answer:

The fifth term of the GP is $\displaystyle{\mathsf{\mathbf{\:\dfrac{1}{8}}}}$

Step-by-step-explanation:

We have given that,

For a GP,

$\displaystyle{\sf\:t_3\:=\:\dfrac{1}{2}}$

The fourth term of the GP is 4 times its sixth term.

We know that,

$\displaystyle{\pink{\sf\:t_n\:=\:a\:r^{n\:-\:1}}}$

$\displaystyle{\implies\sf\:t_3\:=\:a\:r^{3\:-\:1}}$

$\displaystyle{\implies\sf\:\dfrac{1}{2}\:=\:ar^2}$

$\displaystyle{\implies\sf\:ar^2\:=\:\dfrac{1}{2}\:\qquad\dots\:(\:1\:)}$

Now,

$\displaystyle{\sf\:t_4\:=\:4\:t_6}$

$\displaystyle{\implies\sf\:a\:r^{4\:-\:1}\:=\:4\:a\:r^{6\:-\:1}}$

$\displaystyle{\implies\sf\:a\:r^3\:=\:4\:a\:r^5}$

$\displaystyle{\implies\sf\:\dfrac{1}{4}\:=\:\dfrac{\cancel{a}\:r^5}{\cancel{a}\:r^3}}$

$\displaystyle{\implies\sf\:r^{5\:-\:3}\:=\:\dfrac{1}{4}}$

$\displaystyle{\implies\sf\:r^2\:=\:\dfrac{1}{4}\:\qquad\dots\:(\:2\:)}$

Now, we have to find the fifth term of the GP.

$\displaystyle{\sf\:t_5\:=\:a\:r^{5\:-\:1}}$

$\displaystyle{\implies\sf\:t_5\:=\:a\:r^4}$

$\displaystyle{\implies\sf\:t_5\:=\:a\:r^2\:\times\:r^2}$

$\displaystyle{\implies\sf\:t_5\:=\:\dfrac{1}{2}\:\times\:\dfrac{1}{4}\:\qquad\dots\:[\:From\:(\:1\:)\:\&\:(\:2\:)\:]}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:t_5\:=\:\dfrac{1}{8}}}}}$

∴ The fifth term of the GP is $\displaystyle{\mathsf{\mathbf{\:\dfrac{1}{8}}}}$

Answer 2

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Answer

• Given, for a GP

t3 = 1 / 2

The fourth term of GP is 4 times its sixth term.

• We know,

tn = a r^n-1

➜t3 = a r^3-1

➜1 / 2 = ar^2

➜ar^2 = 1 / 2 ----> (1)

• Now,
• t4 = 4 t6

➜ar^4-1 = 4ar^6-1

➜ar^3 = 4ar^5

➜ 1/4 = ar^5 / ar^3

➜r^5-3 = 1 / 4

➜r^2 = 1 / 4 ----> (2)

• Now we have to find the 5th term of GP..

t5 = ar^5-1

➜t5 = ar^4

➜t5 = ar^2 × r^2

➜t5 = 1/2 × 1/4 (from eqn 1 & 2)

➜t5 = 1/8

The 5th term of GP is 1/8