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

" ... the sum of the first five terms is one-fourth of the next terms."

confusing, I will assume you meant:
the sum of the first five terms is one-fourth of the next term.
i.e.
the sum of the first five terms is one-fourth of the 6th terms.

a = 2
6th term = a+5d
sum(5) = (5/2)(2a + 4d) = (5/2)(4 + 4d)
= 10 + 10d

10 + 10d = (1/4)(a+5d)
40 + 40d = 2 + 5d
35d = -38
d = -38/35

term(12) = a + 11d
= 2 - 11(-38/35) ≠ -112

check the wording of your question.

working backwards:
if a = 2 and term(20) = -112
term(20) = a+19d
2 + 19d = -112
19d = -114
d = -6

so the sum of the first five terms would be
(5/2)(4 + 4(-6)) = -50
Now that is supposed to be 1/4 of "something"

To solve this problem, we need to first find the common difference (d) of the arithmetic progression (A.P.).

We are given that the first term (a) is 2.

Next, we know that the sum of the first five terms of the A.P. (S₅) is equal to one-fourth of the sum of the next terms. Mathematically, this can be written as:

S₅ = 1/4 * Sₙ

To find S₅, we use the formula for the sum of the first n terms of an A.P.:

Sₙ = (n/2) * (2a + (n-1)d)

Plugging in the given values, we have:

S₅ = (5/2) * (2(2) + (5-1)d)
S₅ = (5/2) * (4 + 4d)
S₅ = 10 + 10d

Now, let's substitute this value into the equation S₅ = 1/4 * Sₙ:

10 + 10d = (1/4) * Sₙ

To solve for d, we need to find Sₙ. The general formula for Sₙ can be rearranged as:

Sₙ = (n/2) * (2a + (n-1)d)

Plugging in the given values, we have:

Sₙ = (n/2) * (2(2) + (n-1)d)
Sₙ = (n/2) * (4 + (n-1)d)
Sₙ = (n/2) * (4 + 4d)

Now, we can substitute this value back into the equation 10 + 10d = (1/4) * Sₙ:

10 + 10d = (1/4) * [(n/2) * (4 + 4d)]

To simplify, let's multiply both sides of the equation by 4:

40 + 40d = n * (4 + 4d)

Now, let's substitute n = 20 (since we want to find the 20th term) and solve for d:

40 + 40d = 20 * (4 + 4d)
40 + 40d = 80 + 80d

Rearranging the equation, we have:

80d - 40d = 80 - 40
40d = 40
d = 1

So, we have found that the common difference (d) is 1.

Now, let's find the 20th term (a₂₀) using the formula for the nth term of an A.P.:

a₂₀ = a + (n-1)d

Plugging in the given values, we have:

a₂₀ = 2 + (20-1) * 1
a₂₀ = 2 + 19
a₂₀ = 21

Therefore, the 20th term (a₂₀) in the A.P. is 21.