# Arithmetic and Multiplicative Progressions

Concept: Identifying the series stated in the question, Summing terms of the identified sequence.

The GMAT Sample Math question given below is from the topic Arithmetic Progressions. Problems on Arithmetic Progressions are quite easy if one understands that the basic concept behind AP is just an extrapolation of simple multiplication tables.

#### Question: What is the sum of all 3 digit numbers that leave a remainder of '2' when divided by 3?

1. 897
2. 164,850
3. 164,749
4. 149,700
5. 156,720

Video explanation will be added soon

#### Identify the series

The smallest 3 digit number that will leave a remainder of 2 when divided by 3 is 101.

The next couple of numbers that will leave a remainder of 2 when divided by 3 are 104 and 107.

The largest 3 digit number that will leave a remainder of 2 when divided by 3 is 998.

It is evident that any number in the sequence will be a 3 digit positive integer of the form (3n + 2).

So, the given numbers are in an Arithmetic sequence with the first term being 101 and the last term being 998 and the common difference being 3.

#### Compute the sum

Sum of an Arithmetic Progression (AP) = $$left[\frac{\text{first term + last term}}{2}\right]n \\$ We know the first term: 101 We know the last term: 998. The only unknown is the number of terms, n. In an A.P., the nth term an = a1 +$n - 1)*d

In this case, therefore, 998 = 101 + (n - 1)* 3

Or 897 = (n - 1) * 3

(n - 1) = 299 or n = 300.

#### Compute the sum

Sum of an Arithmetic Progression (AP) = $$left[\frac{\text{first term + last term}}{2}\right]n \\$ We know the first term: 101 We know the last term: 998. The only unknown is the number of terms, n. In an A.P., the nth term an = a1 +$n - 1)*d

In this case, therefore, 998 = 101 + (n - 1)* 3

Or 897 = (n - 1) * 3

(n - 1) = 299 or n = 300.

Sum of the AP will therefore, be $\left[\frac{101 + 998}{2}\right] * 300 \\$ = 164,850

Choice B is the correct answer.

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