Find the sums given below :
(i) 7 + 10 ½ + 14 +... + 84
(ii) 34 + 32 + 30 +... + 10
(iii) - 5 + (- 8) + (- 11) +... + (- 230)
Solution:
The sum of the first n terms of an Arithmetic Progression is given by Sₙ = n/2 [2a + (n - 1) d] or Sₙ = n/2 [a + l], and the nth term of an AP is aₙ = a + (n - 1)d
Here, a is the first term, d is the common difference and n is the number of terms and l is the last term.
(i) 7 + 10 ½ + 14 + .... + 84
Given,
- First-term, a = 7
- Common Difference, d = 10 ½ - 7 = 21/2 - 7 = 7/2
- Last term, l = 84
To find the total number of terms, we will use the formula a + (n - 1) d.
l = aₙ = a + (n - 1) d
84 = 7 + (n - 1) 7/2
77 = (n - 1) 7/2
22 = n - 1
n = 23
Hence, n = 23
Sₙ = n/2 [a + l]
S₂₃ = 23/2 [7 + 84]
= (23/2) × 91
= 2093/2
= 1046 ½
(ii) 34 + 32 + 30 + .... + 10
Given,
- First term, a = 34
- Common Difference = 32 - 34 = -2
- Last term, l = 10
To find the total number of terms, we will use the formula aₙ = a + (n - 1) d.
l = a + (n - 1) d
10 = 34 + (n - 1) (- 2)
n - 1 = 12
n = 13
Hence, n = 13
Sₙ = n / 2 [a + l]
S₁₃ = 13/2 [34 + 10]
= (13/2) × 44
= 13 × 22
= 286
(iii) - 5 + (- 8) + (- 11) + .... + (- 230)
Given,
- First-term, a = -5
- Common Difference, d = (- 8) - (- 5) = - 8 + 5 = - 3
- Last term, l = - 230
To find the total number of terms, we will use the formula a + (n - 1) d.
l = a + (n - 1)d
- 230 = (- 5) + (n - 1)(- 3)
n - 1 = 225/3
n = 75 + 1
n = 76
Hence, n = 76
Sₙ = n/2 [a + l]
= 76/2 [(- 5) + (- 230)]
= 38 [- 235]
= - 8930
☛ Check: NCERT Solutions for Class 10 Maths Chapter 5
Video Solution:
Find the sums given below : (i) 7 + 10 ½ + 14 +... + 84 (ii) 34 + 32 + 30 +... + 10 (iii) - 5 + (- 8) + (- 11) +... + (- 230)
Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.3 Question 2
Summary:
The sum of the following A.P. terms is as follows, for the first A.P. , 7 + 10 ½ + 14 +... + 84, the sum is 1046.5, for the second A.P., 34 + 32 + 30 +... + 10, the sum is 286, for the third A.P., - 5 + (- 8) + (- 11) +... + (- 230), the sum is - 8930.
☛ Related Questions:
- In an AP (i) Given a = 5, d = 3, an = 50 , find n and Sn. (ii) Given a = 7, a13 = 35 , find d and S13.
- How many terms of the AP. 9, 17, 25 ... must be taken to give a sum of 636?
- The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- The first and the last term of an AP are 17 and 350 respectively.If the common difference is 9,how many terms are there and what is their sum?
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