Question

the sum of n terms of the G.P., 4 + 4√2 + 8+ ……… is 28 + 12√2
; find n.​

Answer 1

Answer:

Number of terms (n) = 5

Step-by-step explanation:

We know that in a GP

S(nth) = a(r^(n) - 1) / (r - 1)

we know that,

a = 4

r = 4√2 ÷ 4 = √2

Thus, r = √2

They have also given us that Sum of these numbers are 28 + 12√2

so, using the formula and putting in the value we know

28 + 12√2 = 4(√2^(n) - 1) / (√2- 1)

(28 + 12√2) × (√2- 1) = 4(√2^(n) - 1)

28√2 - 28 + 24 - 12√2 = 4(√2^(n) - 1)

16√2 - 4 = 4(√2^(n) - 1)

(4(4√2 - 1))/4 = (√2^(n) - 1)

(4√2 - 1) = (√2^(n) - 1)

4√2 - 1 + 1 = √2^(n)

4√2 = √2^(n)

Dividing both sides by √2 we get

(4√2)/√2 = (√2^(n))/√2

4 = √2^(n - 1)

2² = √2^(n - 1)

we know that,

2 = √2 × √2 = (√2)²

so, ((√2)²)² = √2^(n - 1)

also,

(a^m)^n = a^(mn)

(√2)⁴ = √2^(n - 1)

Now the bases are same so,

4 = n - 1

n = 4 + 1 = 5

Thus, the number of terms (n) added here is 5

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Hope it helped and you understood it........All the best