The sum of the series 31 + 33 +.......+ 53 is
504
Given:
The series 31 + 33 +...........+ 53 is an arithmetic series.
Concept:
The nth term of an arithmetic series, an = a1 + (n - 1)d
The sum of the series, Sn = n/2[2a1 + (n-1)d]
Calculation:
⇒ Let the first term of the arithmetic series, a1 = 31
⇒ The second term of the arithmetic series, a2 = 33
And the common difference d, d = a2 - a1 = 33 - 31 = 2
⇒ Therefore, d = 2
⇒ The nth term of the arithmetic series, an = a1 + (n - 1)d
⇒ 53 = 31 + (n - 1)(2)
⇒ 53 - 31 = 2n - 2
⇒ 22 = 2n - 2
⇒ 22 + 2 = 2n
⇒ 2n = 24
⇒ n = 24/2
⇒ n = 12
⇒ Therefore, 53 is the 12th term of the arithmetic series.
⇒ The sum of the series is, Sn = n/2[2a1 + (n - 1)d]
⇒ S12 = (12/2)[2 × 31 + (12 - 1)2]
⇒ 12 × (2/2) [31 + 11]
⇒ 12(42)
⇒ 504
Thus, the sum of the series 31 + 33 +......+ 53 is 504.
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