XAT 2015 Quant & DI: Sum of AP
This question appeared as a part of the quantitative reasoning and data interpretation section of the XAT 2015. A total of 31 questions appeared in this section in XAT 2015. This question is a relatively very easy question and tests the concept of summation of an arithmetic sequence.
Question
What is the sum of the following series? 64, 66, 68, ..... , 100
 1458
 1558
 1568
 1664
 None of the above
Correct Answer Choice B. The sum is 1558.
Explanatory Answer

Step by step solution
The sequence is 64, 66, 68, ..... 100.
The given set of numbers are in an arithmetic progression. Key Data: First term is 64. The common difference is 2. The last term is 100.
 Sum of the first n terms of an AP = \\frac{n} {2} [{2{a_1} + ({n  1}) d}] \\)
To compute the sum, we know the first term a_{1} = 64 and the common difference d = 2.
We do not know the number of terms n. Let us first compute the number of terms and then find the sum of the terms.  Step to compute number of terms of the sequence
a_{n} = a_{1} + (n  1)d
100 = 64 + (n  1)(2)
Therefore, n = 19.
 Sum S_{n} = \\frac{19} {2} [{2({64}) + ({19  1}) (2)}] \\)
S_{n} = \\frac{19} {2} [{128 36}] \\)
S_{n} = 19 * (82) = 1558
The correct answer is Choice B.
 Key Data: First term is 64. The common difference is 2. The last term is 100.

Alternative Method
The sum of any set of numbers = Average of the numbers * number of terms
 Step 1: Compute Average
Average of the terms of an arithmetic sequence = \\left[\frac{first \space term \space + \space last \space term} {2} \right]\\)
Average of this sequence = \\left[\frac{64 \space 100} {2} \right]\\) = 82
 Step 2: Compute Sum Sum = Average * number of terms
We have computed the number of terms in the text book approach. Number of terms = 19.
Therefore, sum = (82) * 19 = 1558
 Step 1: Compute Average

Formulae to remember in Arithmetic Sequence
An arithmetic progression is such a sequence in which every subsequent term of the sequence is obtained by adding the preceding term with a constant number. This constant number is called the common difference(d) of the sequence.
e.g., 2, 4, 6, 8, 10 is an arithmetic sequence.
The second term is obtained by adding the first term by 2. The third term is obtained by adding the second term by 2 and so on.Formula 1: nth term of a Arithmetic Progression a_{n} = a_{1} + (n  1)d, where a is the first term of the sequence and d is the common difference.
Formula 2: Sum of the first n terms of a arithmetic progression =
Alternatively, you can find the sum using this formula S_{n} =
Video Explanation
More Questions on Arithmetic, Geometric Progression
 Find the sum of terms of a GP
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 Sum of AP having ve common difference
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 CAT 2003: Find sum of terms of an AP
 AP: Integers divisible by 4 or 9
 TANCET 2013: Sum of a GP
 XAT 2014: AP & Divisibility
 XAT 2015: Sum of an AP
 Logarithm terms in an AP
 Speed Distance Time & AP
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