Question 35 of 44intermediate🔧 ApplyShort Answer4 marks

If the sum of the first n terms of an AP is 4n - n², what is the first term (that is S₁)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

Correct Answer

(first_term) 3

(nth_term) 5 - 2n

(second_term) 1

(sum_of_first_two_terms) 4

(tenth_term) -15

(third_term) -1

Exercise: EXERCISE 5.3 | Q: 11 | (Page Page 21)
For More Understanding

Explanation

This question tests the relationship between the sum of first n terms (Sₙ) and the nth term (aₙ) of an AP. The key formula from the textbook is: aₙ = Sₙ - Sₙ₋₁. This means the nth term equals the difference between the sum of first n terms and the sum of first (n-1) terms.

Solution Steps

  1. Step 1: Find the first term (S₁) Given: Sₙ = 4n - n² Substituting n = 1: S₁ = 4(1) - (1)² = 4 - 1 = 3 Therefore, the first term a₁ = 3.

  2. Step 2: Find the sum of first two terms Substituting n = 2: S₂ = 4(2) - (2)² = 8 - 4 = 4

  3. Step 3: Find the second term Using aₙ = Sₙ - Sₙ₋₁ a₂ = S₂ - S₁ = 4 - 3 = 1

  4. Step 4: Find the third term S₃ = 4(3) - (3)² = 12 - 9 = 3 a₃ = S₃ - S₂ = 3 - 4 = -1

  5. Step 5: Find the 10th term S₁₀ = 4(10) - (10)² = 40 - 100 = -60 S₉ = 4(9) - (9)² = 36 - 81 = -45 a₁₀ = S₁₀ - S₉ = -60 - (-45) = -15

  6. Step 6: Find the nth term aₙ = Sₙ - Sₙ₋₁ Sₙ = 4n - n² Sₙ₋₁ = 4(n-1) - (n-1)² = 4n - 4 - (n² - 2n + 1) = -n² + 6n - 5 aₙ = (4n - n²) - (-n² + 6n - 5) = 4n - n² + n² - 6n + 5 = 5 - 2n

PreviousNext