Question 34 of 44intermediate🔧 ApplyShort Answer6 marks
Show that a₁, a₂, . . ., aₙ, . . . form an AP where aₙ is defined as below. Also find the sum of the first 15 terms in each case.
(i)
aₙ = 3 + 4n
Answer
The sequence forms an AP with first term a = 7, common difference d = 4, and sum of first 15 terms = 525.
(ii)
aₙ = 9 - 5n
Answer
The sequence forms an AP with first term a = 4, common difference d = -5, and sum of first 15 terms = -465.
Explanation
To show that a sequence forms an AP, we need to verify that the difference between consecutive terms is constant. For aₙ = 3 + 4n, we find aₙ₊₁ - aₙ = 4 (constant). For aₙ = 9 - 5n, we find aₙ₊₁ - aₙ = -5 (constant). Once confirmed as AP, we use the sum formula S = n/2[2a + (n-1)d] from the textbook to find the sum of first 15 terms.