# Removing an increasing amount of marbles per day how many left after 14 days?

#### wduk

##### New member
Hello

I am stuck trying to understand how to solve this.

If i have 5320 marbles, and on the first day i remove 3 marbles. Then for each day after i remove +5 more than the previous day. Say day two would i remove 3+5 marbles, 3rd day i remove 8 + 5 marbles.

So how would i go about writing the sequence for this? It doesn't seem like a geometric or an arithmetic sequence since the amount i remove from the total is not a constant, it is increasing by 5 each time..

Additionally how would i go about calculating how many marbles i would have left on the 14th day?

#### mmm4444bot

##### Super Moderator
Staff member
If i have 5320 marbles, and on the first day i remove 3 marbles. Then for each day after i remove +5 more than the previous day. Say day two would i remove 3+5 marbles, 3rd day i remove 8 + 5 marbles.

So how would i go about writing the sequence for this? It doesn't seem like a geometric or an arithmetic sequence since the amount i remove from the total is not a constant, it is increasing by 5 each time.
Here is a partial list of the numbers of marbles removed (in order, starting with day 1):

3, 8, 13, 18, 23, ...

Can you explain why this does not seem like an arithmetic sequence to you? :cool:

#### wduk

##### New member
Mainly because from my understanding of the rule, is the number must be a constant, but the first amount taken is 3 then from that point on its (n-1)+5 so its not constant, so 3 and 5 are not the same its not a constant?

Or am i confused?

#### mmm4444bot

##### Super Moderator
Staff member
It's the difference between numbers that must be the same.

An arithmetic sequence is an ordered list of numbers where the difference between consecutive elements is constant.

For example: 5, 7, 9, 11, 13, 15, ... is an arithmetic sequence. Every adjacent pair of numbers differ by a constant amount: 2

3, 8, 13, 18, 23, ... is also an arithmetic sequence because the difference between consecutive elements is always 5.

If you're still not sure about the definition of an arithmetic sequence, look up the definition in your textbook and work through the given examples. Alternatively, you can google for video examples on the Internet.

Once you understand the definition, please try your exercise again. :cool:

#### wduk

##### New member
Okay i am still a little bit stuck.

I am doing the sum of the finite arithmetic series here. So i have:

1/2 * 14 (2 * 5320 + ( 14-1)*(-3)) = 74025

But i am not sure on two things, is the first term from my problem the initial number or the number after the first subtraction when calculating the sum of the finite series?

Secondly if it is actually the second number aka 5317 is the number of terms also 1 less so 13 not 14 ?

#### wduk

##### New member
Well then i get:

1/2 * 14(2 * 5306 + (14-1)*(-3)) = 74011

5320 - 74011 = -68691

Though not sure i should be doing -3

#### mmm4444bot

##### Super Moderator
Staff member
is the first term from my problem the initial number or the number after the first subtraction when calculating the sum of the finite series?
The first term of the arithmetic sequence is 3.

You're not going to do anything like "first subtraction, second subtraction, etc." You're going to sum all of the subtracted numbers first (using the formula for the sum of a finite arithmetic sequence), so that you can subtract them all in one step.

#### mmm4444bot

##### Super Moderator
Staff member
Well then i get:

1/2 * 14(2 * 5306 + (14-1)*(-3))
You're close. :cool:

The red value is supposed to be the first term of the sequence: 3

The blue value is supposed to be the common difference in the sequence: 5

#### wduk

##### New member
Ohhhhh now that makes much more sense! Thanks for the help!

#### HallsofIvy

##### Elite Member
Another way of looking at an arithmetic sequence is that the average of all the numbers in the sequence is the same as the average of the first and last numbers in the sequence.

Here, the sequence is 3, 8, 13, ..., 3+ 13(5)= 68. The average of all those numbers is (3+ 68)/2= 35.5 so the sum of all 14 is 14(35.5)= 497. 497 marbles were removed leaving 5320- 497= 4823.