Question

If the sum of the first n term of an AP is 4n - n2 What is the fifth term of AP?​

Answer 1

ATQ, Sum of the first n terms of an AP is 4n - n².

⇒ S$\sf _{n}$ = 4n - n²

Let n = 1.

⇒ S₁ = 4(1) - (1)²

⇒ S₁ = 4 - 1

⇒ S₁ = 3

⇒ S$\sf _{n}$ = 4n - n²

Let n = 2.

⇒ S₂ = 4(2) - (2)²

⇒ S₂ = 8 - 4

⇒ S₂ = 4

⇒ S$\sf _{n}$ = 4n - n²

Let n = 3.

⇒ S₃ = 4(3) - (3)²

⇒ S₃ = 12 - 9

⇒ S₃ = 3

Therefore:

⇒ First term (a₁) = S₁

(Because the Sum of the first term is the same as the first term)

⇒ a₁ = 3

⇒ Second term (a₂) = S₂ - S₁

⇒ a₂ = 4 - 3

⇒ a₂ = 1

⇒ Common difference = a₂ - a₁

⇒ Common difference = 1 - 3

⇒ Common difference = -2

Now that we have the first term and the common difference, let's find the fifth term.

We know that,

⇒ a$\sf _{n}$ = a + (n - 1)d

Here, n = 5. (Position of the term)

⇒ a₅ = 3 + (5 - 1)(-2)

⇒ a₅ = 3 + (4)(-2)

⇒ a₅ = 3 - 8

⇒ a₅ = -5

Therefore the 5th term of this AP is -5

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Definition of the terms used:

a ➞ First term of the Arithmetic Progression.

d ➞ Common difference of the AP.

a$\sf _{n}$ ➞ Position of the term.

S$\sf _{n}$ ➞ Sum of the first n terms.