What is the value of n if the sum of the consecutive odd intergers from 1 to n equals 169?

A) 47
B) 25
C) 37
D) 33
E) 29

Sum of the odd consecutive integers from 1 to n, where the # of terms is N : \(N^2\).

Thus, \(N^2 = 169\): N=13.

Thus, there are 13 terms in the series, and 13th term :First term+(# of terms-1)*Common difference : 1+(13-1)*2 = 25

B.
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Before you tackle this question you must first understand that the question is comprised of two key parts, 1st is finding out how many terms is in that sequence and 2nd what actual number value that term is. In an arithmetic progression, in this case consecutive odd integers 1, 3, 5, ...., there are two set of rules.

Rule #1 (Arithmetic Sequence): xn = a + d(n-1) Identifies what the actual # in the sequence would be. Each number in the sequence has a term such as 1(is the first term), 3(is the second term) and so on. So if I were to ask you to find out what the 10th term is of that sequence you would use that formula to find that value.
a=1 (first term)
d=2 (the common difference) remember in the sequence 1, 3, 5, 7 the common difference is always 2

*On a side note we use n-1 because we don't have d in the first term, therefore if we were solving for the first term we would get 0 as n-1 and 0 times d would give us 0, leaving only the first term. This works regardless what your first term is in any sequence.

But remember the question asks " What is the value of n if the sum of the consecutive odd integers from 1 to n equals 169?" which means we first need a consecutive sequence that sums up to 169 and than find what the value of the n is, in this case it would be the last number in that sequence. In order to find that we first need to know how many terms (how many of the n there is) in order to be able to plug n in this formula given we know what the sum is. For that to happen we need to use Rule #2.

Rule #2 (Summing an arithmetic series): 169 = n/2(2a+(n-1)d). Given the question gives us what the sum is (169 in this case) we would simply use this formula to solve for n. Once we solve for n (13 in this case) we can simply plug n into the first formula (rule 1) and find the value.

It feels very confusing and difficult at first, but once you identify the steps all you need to do is plug and play. We have the sum (169) of a sequence, the number of terms in that sequence is (unknown). Rule #2 tells us how many numbers there are in that sequence and Rule #1 gives us what that last term is.

Since sum = average x quantity:

169 = (n + 1)/2 x [(n -1)/2 + 1]

169 = (n^2 - 1)/4 + (n + 1)/2

Multiplying the equation by 4, we have:

676 = n^2 - 1 + 2(n + 1)

676 = n^2 - 1 + 2n + 2

n^2 + 2n - 675 = 0

(n - 25)(n + 27) = 0

n = 25 or n = -27

Since n can’t be negative, n = 25.

Answer: B
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