writing algebraic equation for: Each time you buy card, price goes up 20 units....

Colt

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[FONT=.SF UI Text][FONT=.SFUIText]Every time you buy a card the price goes up 20 units. The first card costs 60 units. How would you write an equation to find the TOTAL cost for X cards bought?[/FONT][/FONT]
[FONT=.SF UI Text][FONT=.SFUIText]1st card costs 60[/FONT][/FONT]
[FONT=.SF UI Text][FONT=.SFUIText]2nd costs 80[/FONT][/FONT]
[FONT=.SF UI Text][FONT=.SFUIText]3rd costs 100[/FONT][/FONT]
[FONT=.SF UI Text][FONT=.SFUIText]So the total cost for 3 cards is 240.[/FONT][/FONT]
[FONT=.SF UI Text][FONT=.SFUIText]I dont know how to turn it into an equation though. If I wanted to go to 60 cards it would take forever.[/FONT][/FONT]
 
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This is a contradiction.

You are correct i should have said the 1st was 60 the second was 80 and the 3rd was 100 so the total for buying all cards would be 240. Still besides that i just really want to know the equation because of a game that i am playing. To know if it is worth buying cards, or if i would even have enough money to buy how many i need. Game money not real life money.
 
The progression of card prices {60, 80, 100, 120, 140,…} is called an Arithmetic Sequence.

There is a formula, to sum the first n elements of an Arithmetic Sequence. This formula requires knowing both the first element and the nth element (i.e., the last element to be added).

The sum of the first n elements = (number of elements being added)*(the average of first and nth elements)

Let symbol Pn = the price of the nth card

In other words:

P1 = 60
P2 = 80
P3 = 100
P4 = 120
et cetera …

With this notation, the formula for the sum of the first n elements is:

Sum = n * (P1 + Pn)/2

There is also a formula for Pn (the price of the nth card):

Pn = 60 + 20(n - 1)

By substituting values for n, you can see that this formula gives the price of each card.

Let's say we buy 50 cards. We already know the price of the 1st card (60 units).

The price of the 50th card is:

P50 = 60 + 20(50 - 1) = 1040

Therefore, the total for buying all 50 cards is:

Sum = 50 * (60 + 1040)/2 = 27500 units :cool:
 
The progression of card prices {60, 80, 100, 120, 140,…} is called an Arithmetic Sequence.

There is a formula, to sum the first n elements of an Arithmetic Sequence. This formula requires knowing both the first element and the nth element (i.e., the last element to be added).

The sum of the first n elements = (number of elements being added)*(the average of first and nth elements)

Let symbol Pn = the price of the nth card

In other words:

P1 = 60
P2 = 80
P3 = 100
P4 = 120
et cetera …

With this notation, the formula for the sum of the first n elements is:

Sum = n * (P1 + Pn)/2

There is also a formula for Pn (the price of the nth card):

Pn = 60 + 20(n - 1)


Thank you so much. I really appreciate it.

Also 60 + 20(n - 1) is the same as 40 + 20n, is it not?
60 + 20(n - 1) = 60 + 20n - 20
If we just add -20 plus 60, it becomes the same thing right? Would the simplified version of the equation not be easier to deal with? Just asking a general question now. You have answered my first question perfectly!
 
… 60 + 20(n - 1) is the same as 40 + 20n, is it not?

… Would the simplified version of the equation not be easier to deal with? …
Yes, 40 + 20n is a simplified version.

Is it easier to use? You tell me. ;) If we mentally subtract 1 from n, then each version requires the same amount of calculator keystrokes. But, there is that mental subtraction …
 
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