# If the sum of the first n terms of an AP is 4n - n^{2}, what is the first term (that is S_{1})? What is the sum of the first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

**Solution:**

The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1) d] or Sₙ = n/2 [a + l], and the nth term of an AP is aₙ = a + (n - 1)d

Here, a is the first term, d is the common difference and n is the number of terms and l is the last term.

Given,

- Sum of first n terms, Sₙ = 4n - n²

Therefore,

Sum of first term, S₁ = 4 × 1 - 1² = 4 - 1 = 3

Sum of first two terms, S₂ = 4 × 2 - 2² = 8 - 4 = 4

Sum of first three terms, S₃ = 4 × 3 - 3² = 12 - 9 = 3

Second term, a₂ = S₂ - S₁ = 4 - 3 = 1

Third term, a₃ = S₃ - S₂ = 3 - 4 = - 1

Tenth term, a₁₀ = S₁₀ - S₉

= (4 × 10 - 10^{2}) - (4 × 9 - 9^{2})

= (40 - 100) - (36 - 81)

= - 60 + 45

= - 15

n^{th} term, aₙ = Sₙ - Sₙ₋₁

= [4n - n²] - [4 (n - 1) - (n - 1)² ]

= 4n - n² - 4n + 4 + (n - 1)²

= 4 - n² + n² - 2n + 1

= 5 - 2n

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 5

**Video Solution:**

## If the sum of the first n terms of an AP is 4n - n², what is the first term (that is S_{1})? What is the sum of the first two terms? What is the second term? Similarly find the 3rd, the 10th and the nth terms

Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.3 Question 11

**Summary:**

If the sum of the first n terms of an AP is 4n - n^{2}, then the first term is equal to 3, the sum of first two terms is equal to 4, the second term is equal to 1, and 3rd term, 10th term and the nth terms are equal to -1, -15, (5 - 2n) respectively.

**☛ Related Questions:**

- Find the sum of the first 40 positive integers divisible by 6.
- Find the sum of the first 15 multiples of 8.
- Find the sum of the odd numbers between 0 and 50.
- A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty if he has delayed the work by 30 days?