Set S consists of 9 consecutive odd integers such

Question

Q59. Set S consists of 9 consecutive odd integers such that the sum of first 5 integers is 155. What is the sum of all the integers in set S?​

A. 309 ​

B. 315​

C. 319​

D. 323​

E. 327​

CrackverbalGMAT · Answer

The integers are odd and positive.

Let them be x, x+2, x+4, x+6, x+8, x+10, x+12, x+14 and x+16

Sum of the first 5 terms = x + x + 2 + x + 4 + x + 6 + x + 8 = 155

5x + 20 = 155

5x = 155 - 20 = 135

x = 27

So the numbers are 27, 29, 31, 33, 35, 37, 39, 41 and 43

There are 3 methods to move ahead here

Method (1)   Simple addition - This is fine in this case as there aren't too many terms. The sum is 315.

Method (2)   Use the concept of Arithmetic Progression. Here the sum of terms is given by  S_n  =  \frac{n}{2}

Where:

n = number of terms = 9

a = first term = 27 and

d = common difference between the terms = 2

Therefore  S_n  =  \frac{9}{2}    =  \frac{9}{2}  * (54 + 16) =  \frac{9}{2}  * (70) = 630/2 = 315

Method 3   requires us to know that the sum of the 1st n odd numbers starting from 1 is  n^2 .

We know the terms range from 27 to 43.

So we need to find the sum of the sum of all odd numbers from 1 to 43 minus all odd numbers from 1 to 25

43 is the (43 + 1) / 2 = 22nd odd number and 25 is the (25+1)/2 = 13th odd number

Therefore the sum =  22^2 - 13^2  = 315

Option B

Arun Kumar

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