How can I find the sum of the first 365 numbers in a sequence?

The specific sequence I was given is 3, 6, 9, 12, 15, ..., increasing by 3 each time. It represents somebody saving pennies every day for a year, putting 3 in the first day, 6 in the second, 9 in the third, and so on. I need to know how much money they will have by the end of a year (not a leap year).

Question

How can I find the sum of the first 365 numbers in a sequence?

The specific sequence I was given is 3, 6, 9, 12, 15, ..., increasing by 3 each time. It represents somebody saving pennies every day for a year, putting 3 in the first day, 6 in the second, 9 in the third, and so on. I need to know how much money they will have by the end of a year (not a leap year).

anonymous · Answer

I've only ever done these problems by actually going through and finding all the terms and manually adding them together, but 365 is a bit much, there's gotta be a faster way to do it

campbell_st · Answer

this is an arithmetic sequence. the formula for the sum of the sequence is 

$$S_{n} = \frac{n}{2}[2a + (n - 1) \times d]$$

n = number of terms in the sequence, in your question n = 365

a = 1st term, in your sequence a = 3

d = common difference, or what is is always increasing by  d = 3

substitute these values into the formula above and you'll get the sum

anonymous · Answer

So it would be 365/2 [2(3)+(365-1)*3

campbell_st · Answer

thats it

anonymous · Answer

I think I got confused when solving it... I got 202020

anonymous · Answer

Is that what I should have gotten?

anonymous · Answer

And if I convert that from pennies to dollars it would become $2,020.20?

campbell_st · Answer

I got 200385

anonymous · Answer

I just calculated it again and got 200,385

anonymous · Answer

Oh lol

campbell_st · Answer

and the conversion is correct...

anonymous · Answer

Alright so it's $2,003.85 (:

anonymous · Answer

Thank you!