The sum of the first n positive odd integers is n^2. What is the sum

Question

The sum of the first n positive odd integers is \(n^2\). What is the sum of the odd integers from 25 to 79, inclusive?

A. \(39^2 - 11^2\)
B. \(39^2 - 12^2\)
C. \(40^2 - 12^2\)
D. \(79^2 - 22^2\)
E. \(80^2 - 23^2\)

gmatophobia · Accepted Answer

The sum of the odd integers from 25 to 79, inclusive = The sum of the odd integers from 1 to 79, inclusive - The sum of the odd integers from 1 to 23, inclusive

Number of integers from from 1 to 79, inclusive =  n_1 = \frac{79-1}{2} + 1 = 40

Number of integers from from 1 to 23, inclusive =  n_2 = \frac{23-1}{2} + 1 = 12

The sum of the odd integers from 25 to 79, inclusive =  n_1^2 - n_2^2 = 40^2 - 12^2

Option C

ENGRTOMBA2018 · Answer

Follow posting guidelines (link in my signatures). Any and all analyses should either be put under spoilers or in the next post.

As for this question, look below.

You can solve the question in 2 ways:

Method 1:  Sum of an arithmetic progression (=a series such as a1 a2 a3 a4 .... such that a1-a2 = a2-a3 = a3-a4 = constant, difference between consecutive terms remains constant.) =

S =  \frac {n }{2} , where n = number of terms, a = 1st term, d = constant difference

For the question asked, odd number from 25 to 79 are 25,27,29,31...77,79 and any 2 consecutive terms differ by 2. Thus, n = (79-25)/2+1 = 28, d = 2, a = 25.

Therefore, Sum = S =  \frac {n }{2}  =  \frac {28 }{2}  =  28*52 =  (40-12)*(40+12)  =  40^2 - 12^2  (as  (a+b)*(a-b) = a^2-b^2 ).

Alternately, Method 2 , you are given that sum of n odd integers =  n^2 .

Thus Sum of odd integers from 25 to 79 = Sum of odd integers from 1 to 79 - Sum of odd integers from 1 to 23 (chose 23 as we need 25 in our final calculations).

==> S =  {n_1}^2 - {n_2}^2 , where  n_1  = 40 and  n_2 = 12 ==> S =  40^2 - 12 ^2 .

Thus C is the correct answer.

mvictor · Answer

oh man..spent some additional seconds because I did not look at the answer choices..
79 is the 40th odd number
to find the sum of the odd numbers from 25 to 79, we need to deduct from the sum of the first 40 odd numbers, the sum of the first 12 odd numbers. note that 25 is the 13th odd number, thus, 23 is the 12.

40^2 - 12^2
i went even further, and expanded it as: (40+12)(40-12)=52x28...

Riya1475828 · Answer

Edit the source to gmat prep focus

Bunuel · Answer

Added the source. Thank you!

kayze · Answer

First find how many odd integers exist between 1 and 79. There are 40 since 79 + 1 = 80, 80/2 = 40

If you scan answer choices, you could immediately choose C due it it being the only choice with  40^2

But continuing on, now find how many odd integers exist between 1 and 23 (since 25 is included in 25 to 79 inclusive). 23 + 1 = 24, 24/2 = 12

So the answer is  40^2 - 12^2

karu26 · Answer

This is can be done by using logic

sum of odd integers between 25 to 79 = Sum of odd integers from 1 to 79 - Sum of odd integers from 1 to 23.

number of odd integers between 1 and 79 = 40
number of odd integers between 1 and 23 = 12

since sum of first n odd positive integers is n^2, the answer should be

40^2 - 12^2 that is option C.

bumpbot · Answer

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The sum of the first n positive odd integers is n^2. What is the sum

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The sum of the first n positive odd integers is n^2. What is the sum

Prep Toolkit

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