What is the Value of k if the Sum of Consecutive Odd Integers From 1 to k Equals 441?

Question: What is the value of k if the sum of consecutive odd integers from 1 to k equals 441?

1. 47
2. 41
3. 37
4. 33
5. 29

“What is the value of k if the sum of consecutive odd integers from 1 to k equals 441?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach 1

Consecutive odd integers represent an evenly spaced set (or arithmetic progression). Now, the sum of the terms in any evenly spaced set is the mean (average) multiplied by the numbers of terms, where the mean of the set is = \(\frac{firstterm+lastterm}{2}=\frac{1+k}{2}\)

Average = \(\frac{firstterm+lastterm}{2}=\frac{1+k}{2}\)

# of the terms =\(\frac{k-1}{2} \) + 1 =\(\frac{k+1}{2}\) (# of terms is an evenly spaced set is \(\frac{lastterm-firstterm}{commomdifference}+1\)

Sum = \(\frac{1+k}{2}\) * \(\frac{k+1}{2}\) = 441

Simplify: \((k+1)^2\) = 4 * 441

So, k + 1 = 2 * 21 = 42, giving k = 41

Correct option: B

Approach 2:

Sum of n evenly spaced numbers = \(\frac{n}{a}[2a+(n-1)d]\)

Where n is the number of odd no.

a = \(1^{st}\) number and d = common difference

Number of integers = \(\frac{(last-first)}{2}+1\)

Number of odd number = \(\frac{k-1}{2}\)

Here, a = 1, d = 2

441 = k + \(\frac{1}{4}\) [1 + (k + \(\frac{1}{2}\) - 1) 2]

441 = k + \(\frac{1}{4}\) [2 + k + 1 -2]

441 * 4 = \((k+1)^2\) = 441

k + 1 = 21 * 2 = 42

k = 41

Correct option: B

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