Question 427328
the multiples of 3 are 3,6,9,12,15,18,21,24,27,30,......... all the way up to 99.


this is an arithmetic series with a common difference of 3.


the formula for the sum of an arithmetic series is:


Sn = (n/2) * (a1 + an)


Sn = the sum of the n terms in the sequence.
a1 = the first term in the sequence.
an = the nth term in the sequence.


our first term is 3 and our last term is 99.


the common difference is 3.


to find the number of terms, we take the last term and divide it by 3 to get 33.


there are 33 terms in the sequence, so n = 33


our formula becomes Sn = (33/2 * (3 + 99) which becomes (33/2) * 102 which becomes 33 * 51 which becomes 1683.


to see how this works, we can use smaller numbers.


assume the first term is 3 and the last term is 9.


the common difference is 3.


the number of terms is 9/3 = 3


the formula states that Sn = (3/2) * (3 + 9) = (3/2) * (12) = 3*6 = 18


if we sum up the terms in the sequence we should equal 18.


3 + 6 + 9 = 9 + 9 = 18.


the formula works.


same thing happens with the larger numbers only it's much harder to show you because there's so many more numbers to add up.