Question

$\sf\implies\:Which \:term\: of\: the\: GP.\:\ =>\ 2,\:8, \:32,\: ……. \:is \:512?$

Answer 1

Answer:

5th term

Step-by-step explanation:

Given: GP = 2, 8, 32, ... 512

To Find: Which term is 512

General Form of a Term in GP = arⁿ⁻¹

Here 'n' refers to the cardinal value of the term. ( Like 1st, 2nd, 3rd, etc. )

⇒ arⁿ⁻¹ = 512

According to the GP,

a = 2, r = 4

Substituting the values we get,

⇒ 2 × 4ⁿ⁻¹ = 512

⇒ 4ⁿ⁻¹ = 512 / 2 = 256

256 can also be written as 4⁴

⇒ 4ⁿ⁻¹ = 4⁴

Since Bases are equal, we can equate the powers. Hence we get,

⇒ n - 1 = 4

⇒ n = 4 + 1

⇒ n = 5

Hence 512 is the 5th term of the GP.

Answer 2

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⇒ given G.P. : 2, 8, 32, .........is 512

a = 2 ; r = a₂ / a₁ = 8 / 2 = 4

let nth term of the G.P. be 512

an = a . r ⁿ⁻¹

512 =2 × (4)ⁿ⁻¹

2⁹ = 2 × (2²) ⁿ⁻¹  [512 can be written as 2⁹]

2⁹ = 2¹  2²⁽ⁿ⁻¹⁾

2⁹ = 2²ⁿ⁻²⁺¹

2⁹ = 2²ⁿ⁻¹

512 = 2⁹

2n-1 = 9 [bases are equal, exponents are also equal]

2n = 9+1

n= 10/2 = 5

Hence , 512 is the 5th term of the given G.P.

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