Question

the first term of G.P is 64 and the 5th term is 4.if sum of all items 128.find the common ratio​

Answer 1

Step-by-step explanation:

$\sf{formula \:of\:nth\:term\:in\:G.P\:is\:a {r}^{n - 1} }$

$\sf{n = 5,nth\:term = 4,a = 64}\\\sf{4 = 64 {r}^{4}\rightarrow \frac{4}{64} = {r}^{4} \rightarrow \frac{1}{16} = {r}^{4} }\\\sf{ (\frac{1}{2})^4 = r^4\rightarrow r = \frac{1}{2} }$

Answer 2

The common ratio (r) will be 1/2.

Given,

First term of GP = a = 64

5th term of GP = 4

Sum of GP = 128

To Find,

Common ratio (r)

Solution,

A Geometric Progression (GP) sequence is a recursive sequence in which each succeeding term is generated by multiplying the preceding term by a constant, which is known as the common ratio.

GP → a, ar, ar², ar³, ..................... arⁿ⁻¹

Now, we have been given 1st and 5th terms,

a₁ = a = 64                 (1)

a₅ = ar⁴ = 4               (2)

Substituting the value of (1) in (2)

64r⁴ = 4

r⁴ = $\frac{4}{64}$

r = $(\frac{4}{64} )^\frac{1}{4}$

r = $(\frac{1}{16} )^\frac{1}{4}$

r = $\frac{1}{2}$

Hence, the common ratio is 1/2.

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