Question

In an A.P. the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20ᵗʰ term is -112.

Answer 1

The common difference is -12, and the 20th term of the AP is -112.

Step-by-step explanation:

Let's denote the common difference of the arithmetic progression (AP) as d.

The first term is given as 2, so the second term can be found by adding the common difference: 2 + d.

Using the arithmetic series formula, the sum of the first five terms (S5) is calculated as:

• S5 = (5/2)(2 + (2 + d) * 5)

We are given that the sum of the first five terms is one-fourth of the next five terms (S10):

• S5 = (1/4)S10

Substituting the values, we get:

• (5/2)(2 + (2 + d) * 5) = (1/4)(10/2)(2 + (2 + d) * 10)

Simplifying this equation gives:

• d = -12

Therefore, the common difference is -12. Using the formula to find the 20th term:

• a20 = a1 + (20 - 1)d = 2 + (19)(-12) = -112.

So, the 20th term of the AP is -112.

Learn more about Arithmetic Progression (AP)