"Odd sum - even sum": How did you get to 61 as the 31st odd number?

Link

https://gre.magoosh.com/questions/213

Question

How did you get to 61?

Answer

We could use a formula if you like memorizing things, but this just relies on a bit of critical thinking. Every time we go to the next even (or odd) number, we are adding 2. That means the nth positive even number is going to be 2 times n. In the list of 2,4,6, and 8, for example, the first number, 2, is 1*2, the second number, 4, is 2*2, the third number, 6, is 3*2, etc.... So we can quickly say that the 30th number will be 30*2 = 60.

The list of odd numbers is very similar, but always 1 less. For example, the fourth number in 1,3,5,7 is (4*2) - 1 = 7. That means the 31st number in a list of odd numbers must be 31*2 - 1 = 62 -1 = 61.

The formula basically does the same thing, but if you're interested here it is:

t = value of nth term in sequence
a = first term in sequence
n = number of terms in sequence
d = difference between terms in sequence

t = a + (n-1)d.

So in the 31st odd number example, we plug in like this:

t = 1 + (31-1)2
t = 1 + (30)2
t = 1 + 60
t = 61

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