An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Find the AP
Solution:
Given, the series consists of 37 terms.
Sum of three middle most terms = 225
Sum of last three terms = 429
We have to find the AP.
Number of terms, n = 37 is odd
Middle term = (n+1)/2 th term
= (37+1)/2
= 38/2
= 19th term.
Three middle most terms will be 18, 19 and 20th term.
The nth term of the series in AP is given by
aₙ = a + (n - 1)d
18th term = a + (18 - 1)d = a + 17d
Similarly, 19th term = a + 18d
20th term = a + 19d
As per given condition, 18th + 19th + 20th term = 225
a + 17d + a + 18d + a + 19d = 225
By grouping,
a + a + a + 17d + 18d + 19d = 225
3a + 54d = 225
Dividing by 3,
a + 18d = 75 -------------------- (1)
Last three terms will be 35th, 36th and 37th terms.
35th term = a + 34d
36th term = a + 35d
37th term = a + 36d
As per given condition, 35th + 36th + 37th terms = 429
a + 34d + a + 35d + a + 36d = 429
By grouping,
a + a + a + 34d + 35d + 36d = 429
3a + 105d = 429
Dividing by 3,
a + 35d = 143 -------------------- (2)
Subtracting (1) and (2),
a + 35d - (a + 18d) = 143 - 75
a + 35d - a - 18d = 68
35d - 18d = 68
17d = 68
d = 68/17
d = 4
Put the value of d in (1),
a + 18(4) = 75
a + 72 = 75
a = 75 - 72
a = 3
The AP in terms of common difference is given by
a, a+d, a+2d, a+3d,......., a+(n-1)d
So, 3, 3+4, 3+2(4), 3+3(4),......
Therefore, the series in AP is 3, 7, 11, 15,.....
✦ Try This: An AP consists of 47 terms. The sum of the three middle most terms is 245 and the sum of the last three is 359. Find the AP
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 5
NCERT Exemplar Class 10 Maths Exercise 5.4 Problem 4
An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Find the AP
Summary:
An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. The AP is 3, 7, 11, 15,.....
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