How many terms of the AP: -15, -13, -11,--- are needed to make the sum -55? Explain the reason for the double answer
Solution:
Given, the series in AP is -15, -13, -11,.......
We have to find the number of terms required to make the sum equal to -55.
From the series,
First term, a = -15
Common difference, d= -13 - (-15) = -13 + 15 = 2
The sum of the first n terms of an AP is given by
Sₙ = n/2[2a + (n-1)d]
So, Sₙ = n/2[2(-15) + (n - 1)(2)]
= n/2[-30 + 2n - 2]
= n/2[2n -32]
= (n/2)2[n - 16]
= n(n - 16)
= n² - 16n
Given, the sum is -55
So, Sₙ = -55
n² - 16n + 55 = 0
On factoring,
n² - 11n - 5n + 55 = 0
n(n - 11)- 5(n - 11) = 0
(n - 5)(n - 11) = 0
Now, n - 5 = 0
n = 5
Also, n - 11 = 0
n = 11
Therefore, the sum of the first 5 terms or the sum of the first 11 terms of an AP is equal to -55.
The 11 terms of the series are -15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5.
The reason for the double answer is that the AP is increasing with positive values.
So, the sum of the first 5 terms will be equal to -55
As the AP increases with positive values the sum of the first 11 terms equals -55, as the last 6 terms sum up to 0.
✦ Try This: How many terms of the AP: 5, 10, 15,.... are needed to make the sum 140
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 5
NCERT Exemplar Class 10 Maths Exercise 5.3 Problem 32
How many terms of the AP: -15, -13, -11,--- are needed to make the sum -55? Explain the reason for the double answer
Summary:
5 or 11 terms of the AP: -15, -13, -11,--- are needed to make the sum -55. The reason for the double answer by taking 11 terms in this AP, the last 6 terms sum up to 0, making the whole sum as -55
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