What is the sum of odd integers from 35 to 85, inclusive?

A

take average and multiply by number of terms

here number of odd terms=26 and average is 60.

answer=1560

Hi Skywalker18,

I do understand the following on this exercise:

1. We're dealing with consecutive integers so I can find the average by: \(\cfrac { First\quad Number+Last\quad Number }{ 2 }\)
2. To find the range in a set of numbers for which both end points are inclusive, we: \(\left( Hightest\quad Number-Lowest\quad Number \right) +1\)

I don't understand why you can simply divide the range by 2 in \(\cfrac { \left( Hightest\quad Number-Lowest\quad Number \right) }{ 2 } +1\) to find the number of odd numbers. Would it be the same if I wanted to find the number of even numbers?

Thanks'

A student of mine, who goes by the handle DoctorDoom on this forum, came up with his own way of solving such problems. I am sharing it with his approval:

\(∑ = nc + \frac{1}{2}ni(n - 1)\)

in which n is number of integers within the set, c is the initial value, and i is the interval or space between successive integers. I worked it out with him using the numbers of the problem at hand:

\(∑ = (26)(35) + \frac{1}{2}(26)(2)((26) - 1)\)

\(∑ = 910 + 26(25)\)

\(∑ = 910 + 650\)

\(∑ = 1,560\)

Checks out. If someone else finds this method useful, then thank DoctorDoom for sharing.

- Andrew

The sum of the odd integers from 35 to 85, inclusive, is:

sum = avg x quantity

sum = (85 + 35)/2 * (85 - 35)/2 + 1

sum = 120/2 * 50/2 + 1

sum = 60 * 26 = 1560

Answer: A

Why can't we apply sum of odd integer formula n^2
So it'd be 85^2 -34^2 = 6069 ???

BrentGMATPrepNow may you please help clarify
thank you in advance

You're referring to:
The sum of the first n positive ODD integers = n²

We can use it here.

From 1 to 85 (inclusive) there are 43 ODD integers.
So the sum 1 + 3 + 5 + 7 + . . . 83 + 85 = 43² = 1849


Of course, this sum also includes the odd integers from 1 to 33
So we must subtract the sum of these numbers from 1849
From 1 to 33 (inclusive) there are 17 ODD integers.
So the sum 1 + 3 + 5 + 7 + . . . 31 + 33 = 17² = 289

So the sum of the odd integers from 35 to 85 (inclusive) = 1849 - 289 = 1560

No of terms in the series = (85-35)/2 +1
N = 25 + 1 = 26
We are adding 1 because 35 is also included in the series.

You can also use the nth term of an AP formula to find the number of terms.

Tn = a + (n-1)*d

where,
Tn is the nth term in the series
a is the first term
n is the number of terms
and, d is the common difference

85 = 35+(n-1)*2

Sum of an AP = n/2 * (1st term + last term)
Sum = 26/2 * ( 35+ 85) = 1560

Thanks,
Clifin J Francis

Solution

Given

• A set of odd integers from 35 to 85, inclusive.

To find

• The sum of the given integers.

Approach and Working out
Since we have to consider only the odd integers, we have the sequence: 35, 37, 39,… , 85.

• The difference between any two consecutive terms = 37 – 35 = 39 – 37 = 2.

Thus, we have an arithmetic progression (AP) with the common difference of 2.
The first term of the sequence = a = 35.
Let 85 be the \(n^{th}\) term of the sequence.
Since the \(n^{th}\) term of an AP can be represented as \(T_n = a + (n-1)d\), where d is the common difference, we have

• 85 = 35 + (n-1)2
• 50 = (n-1)2
• 25 = n – 1
• n = 26.

Thus, there are 26 odd integers between 35 and 85 (both inclusive).
The given sequence is an evenly spaced sequence. Thus, the mean of the sequence = average of the first and the last term.
Thus, sum of the terms of the sequence divided by 26 = average of 35 and 85

• \(\frac{Sum}{26} = \frac{(35 + 85)}{2}\)
• \(Sum = \frac{26(120)}{2} = 1,560\)

Correct Answer: Option A

I tried to solve for the number of terms by using AP method -

a = first term + (n-1) x d

But this gives me the value of n as 25 rather than 26, where am I going wrong here?

archit GMATBusters Skywalker18

You are doing some mistake.

since the ques is talking about consecutive odd integers, common difference = 2

85 = 35 + (n-1)*2
50= (n-1)*2
25 = (n-1)
26 = n

How did you know to use that formula to find the number of odd numbers. what if it was 35 to 86? how do you know when to add the one vs not to add it

Asked: What is the sum of odd integers from 35 to 85, inclusive?

Sum = 35 + 37 + 39 + .... + 85
n = (85-35)/2 + 1 = 26
a = 35
l = 85
d = 2

S = n/2 {2a + (n-1)d} = 26/2 {70 + 25*2} = 13 * 120 = 1560

IMO A

Hi sa800
Thanks for your query.


To get a better understanding of ‘when to add 1’ and ‘when not to add 1’, let me show you a few examples related to the same concept. After looking at these examples, you will be able to see the relevance of adding 1. In fact, you will see we always add 1, and it’s actually some other terms that need to be taken care of.


So, without much ado, let’s LEARN THROUGH EXAMPLES!


EXAMPLE 1:
Find the number of odd integers from 34 and 85, inclusive.

Observe that even though the smallest integer in the given range is 34, the smallest integer that we will consider is 35 (since 34 is even and 35 is the smallest odd integer). Let’s call 35 the first meaningful number – this is the first number that satisfies what we want: an odd integer.

So, the list of odd integers in our range is 35, 37, 39, 41, …, and 85. (Here, 85 is included because the question mentions “inclusive”. We would also have included 34 had it been an odd integer!)

• In the list 35, 37, 39, …, 85, we have:
  
  • First meaningful number = 35 and last meaningful number = 85.
• So, our question boils down to finding the number of odd integers starting from 35 and ending at 85. (Including BOTH)
• The formula we will use is this:
  
  \(\frac{(Last meaningful number – first meaningful number)}{2} + 1\)
• Using this, the answer to this example is = \(\frac{(85 – 35)}{2} + 1\) = 26

EXAMPLE 2:
Find the number of odd integers from 34 to 86, inclusive.

This time also, the first meaningful number = 35 (34, though the smallest, is not odd; the first odd number after 34 is 35.)
Similarly, the last meaningful number = 85 (86, though the largest, is not odd; the largest odd number before 86 is 85.)
So, the list of odd integers in our range is again 35, 37, 39, 41, …, and 85. Hence, the number of odd integers between 34 and 86 is the same as in Example 1 (Ans = 26).

IMPORTANT: Observe how we added 1 in both examples. What varied was not the 1 but the elements that became our first and last meaningful elements.


EXAMPLE 3:
Find the number of odd integers from 33 to 86, inclusive.

This time, the smallest number in the range is also the smallest meaningful number! This is because 33 itself is the smallest odd integer in the range.
But even though 86 is the largest integer in the given range, it is not the largest odd integer. The largest odd integer in the range is 85 and thus, this is the largest meaningful number for us.
So, the list of odd integers in our range is 33, 35, 37, …, and 85.

• Using the formula we discussed, the answer to this example is = \(\frac{(85 – 33)}{2} + 1\) = 27.

TAKEAWAY:
In each example, we began by finding the first and the last term of the required list from within the given range. For convenience, we called these terms “meaningful” terms. And after this, it was always the same formula! 😊



Hope this helps!

Best,
Aditi Gupta
Quant expert, e-GMAT

does this works also for even integers, and also all the intergers between 35 to 85?

let's say the question was: what is the sume of all the even integer between 35 and 85 incuded?
or: what is the sum of all the integers between 35 to 85 included ?

how woud we solve those 2 questions?

Thanks !!!

Where does that rule for the sum comes from?
Like how do i know that i have to add both number then divide by 2, then after time the number of integer?

Quantity = largest number - smallest number + 1 = 26
Avg = (largest number + smallest number) / 2 = 60

Multiply both = 1560

Answer A­

­What else does this rule apply to? Say if they said every 3rd interger, would it be 35 +85 /3? 
Or if they said every interger divsible by 4, then would we start and end with the lowest /highest number?