The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms
Solution:
Given, the ratio of 11th term to the 18th term of an AP is 2:3
We have to find
A) the ratio of the 5th term to the 21st term.
B) the ratio of the sum of the first five terms to the sum of the first 21 terms.
The nth term of the series in AP is given by
aₙ = a + (n - 1)d
11th term = a + (11-1)d = a + 10d
18th term = a + 17d
As per given condition, 11th term/18th term = 2/3
(a+10d)/(a+17d) = 2/3
3(a+10d) = 2(a+17d)
3a + 30d = 2a + 34d
By grouping,
3a - 2a = 34d - 30d
a = 4d ------------------------ (1)
A) to find the ratio of the 5th term to the 21st term
5th term = a + 4d
21st term = a + 20d
(a + 4d) /(a + 20d)
Substitute (1) in the above expression,
(4d+4d)/(4d+20d)
= 8d/24d
= 1/3
Therefore, the ratio of the 5th term to the 21st term is 1:3
B) the ratio of the sum of the first 5 terms to the sum of the first 21 terms.
The sum of the first n terms of an AP is given by
Sₙ = n/2[2a + (n-1)d]
The sum of first five terms is S₅ = 5/2[2a + (5 - 1)d]
S₅ = 5/2[2a + 4d]
S₅ = (5/2)2[a + 2d]
S₅ = 5a + 10d
The sum of first 21 terms is S₂₁ = 21/2[2a + (21 - 1)d]
S₂₁ = 21/2[2a + 20d]
S₂₁ = (21/2)2[a + 10d]
S₂₁ = 21a + 210d
As per given condition, S₅/S₂₁
= (5a+10d)/((21a+210d)
Substitute (1) in the above expression,
= (5(4d) + 10d)/(21(4d) + 210d)
= (20d+10d)/(84d+210d)
= 30d/294d
= 5/49
Therefore, the ratio of the sum of the first 5 terms to the sum of the first 21 terms is 5:49
✦ Try This: The ratio of the 10th term to the 14th term of an AP is 2 : 3. Find the ratio of the 4th term to the 11st term, and also the ratio of the sum of the first five terms to the sum of the first 18 terms
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 5
NCERT Exemplar Class 10 Maths Exercise 5.4 Problem 6
The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms
Summary:
The ratio of the 11th term to the 18th term of an AP is 2 : 3. The ratio of the 5th term to the 21st term is 1:3 and also the ratio of the sum of the first five terms to the sum of the first 21 terms is 5:49
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