Give the value of a1 for the geometric sequence

Question

a₈ = 384

a₁₂ = 6144

Answer 1

a₁₂ = a₈(r)⁴

6144 = 384r⁴

16 = r⁴

±2 = r

a₈ = a₁(2)⁷

384 = a₁(128)

3 = a₁

Answer 2

a₈ = a₁r⁷, a₁₂ = a₁r¹¹;

We have: a₁r⁷ _{= 384 (1);} a₁r¹¹ = 6144 (2).

Deviding (2) by (1) we get r⁴ = 6144/384; r⁴ = 16, r = ± (16)^1/4 = ± 2.; a₁ = 384/r⁷ = 384/(±2)⁷ = ±3

Answer: a₁ = 3 or a₁ = - 3

Answer 3

Thinking about the sequence, we can write the terms like this:

Each term gets multiplied by "r" to get to the next term, So to get from term a₈ to a₁₂, we need to multiply 384 by r⁴ meaning we can write:

384r⁴ = 6144 then solve for r to get:

r⁴ = 16

r = 2

The implicit equation for the sequence will then be a_n = a₀•(2)ⁿ where a₀ is the "zeroth term" (not a "real" term but useful in defining the equation in terms of an exponential equation in "standard form")

Please note that many books teach a version of this which raises "r" to the "n - 1" power but I prefer this methodology because it capitalizes on what you already know about exponential functions.

To find a₀, we can plug in n = 8 when a_n = a₈ = 384 giving us:

384 = a₀•(2)⁸

384 = a₀•(256)

a₀ = 384/256 = 1.5 (making the implicit equation a_n = (1.5)•(2)ⁿ

That mean a₁ = 1.5•2 = 3