List the first five terms of the sequence. a_n = (-1)^{n - 1} /4ⁿ.

Solution:

Given the f(n) = a_n = (-1)^{n - 1} /4ⁿ.

Let us take n = 1, 2, 3, 4, 5

When n = 1 ⇒ f(1) = a₁ = (-1)^{1 - 1} / 4¹ = (-1)⁰ / 4 = 1/4 [ since any number power zero is 1]

When n = 2 ⇒ f(2) = a₂ = (-1)^{2 - 1} / 4² = -1/16

When n = 3 ⇒ f(3) = a₃ = (-1)^{3 - 1} / 4³ = 1/64

When n = 4 ⇒ f(4) = a₄ = (-1)^{4 - 1} / 4⁴ = -1/256

When n = 5 ⇒ f(5) = a₅ = (-1)^{5 - 1} / 4⁵ = 1/1024

Since, even powers of -1 give ‘1’ and odd powers of -1 give ‘-1’.

Therefore, the first five terms of the sequence are 1/4, -1/16, 1/64, -1/256, 1/1024

List the first five terms of the sequence. a_n = (-1)^{n - 1} /4ⁿ.

Summary:

The first five terms of the sequence. a_n = (-1)^{n - 1} /4ⁿ are 1/4, -1/16, 1/64, -1/256, 1/1024.