THE RATIO OF 11th TERM TO 8th TERM OF AN A P IS 2:3 FIND THE RATIO OF THE - Maths - Arithmetic Progressions

Question

THE RATIO OF 11th TERM TO 8th TERM OF AN A P IS 2:3 FIND THE
RATIO OF THE 5th TERM TO TH21th TERM AND ALSO THE RATIO OF THE SUM
OF THE FIRST FIVE TERMS TO THE SUM OF THE FIRST 21 TERMS

Aakash Sharma · Accepted Answer

Let a and d be the first term and common difference of the A.P. respectively.

n^th term of A.P. a_n= a + (n – 1)d

Given, a₁₁ : a₈ = 2 : 3

$∴ \frac{a + (11 - 1) d}{a + (8 - 1) d} = \frac{2}{3} \Rightarrow \frac{a + 10 d}{a + 7 d} = \frac{2}{3} \Rightarrow 3 a + 30 d = 2 a + 14 d \Rightarrow 3 a - 2 a = 14 d - 30 d \Rightarrow a = - 16 d ... (1) ∴ \frac{a_{5}}{a_{21}} = \frac{a + (5 - 1) d}{a + (21 - 1) d} = \frac{- 16 d + 4 d}{- 16 d + 20 d} = \frac{- 12 d}{4 d} = - 3 [Using (1)]$

$\frac{S_{5}}{S_{21}} = \frac{\frac{5}{2} [2 a + (5 - 1) d]}{\frac{21}{2} [2 a + (21 - 1) d]} = \frac{5 (- 32 d + 4 d)}{21 (- 32 d + 20 d)} = \frac{5 \times (- 28 d)}{21 \times (- 12 d)} = \frac{35}{63} {Using (1) and S_{n} = \frac{n}{2} [2 a + (n - 1) d]}$

THE RATIO OF 11th TERM TO 8th TERM OF AN A.P IS 2:3. FIND THE RATIO OF THE 5th TERM TO TH21th TERM AND ALSO THE RATIO OF THE SUM OF THE FIRST FIVE TERMS TO THE SUM OF THE FIRST 21 TERMS .