XAT 2015 Quant & DI: Sum of AP

Step by step solution

The sequence is -64, -66, -68, ..... -100.

The given set of numbers are in an arithmetic progression.

1. Key Data: First term is -64. The common difference is -2. The last term is -100.
   
   
2. 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 a₁ = -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.
   
   
3. Step to compute number of terms of the sequence
   a_n = a₁ + (n - 1)d
   -100 = -64 + (n - 1)(-2)
   Therefore, n = 19.
   
   
4. 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.

Alternative Method

The sum of any set of numbers = Average of the numbers * number of terms

1. 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
   
   
2. 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
   
   

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 = a₁ + (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 =