The arithmetic progression is such sequence which has each her member, since the second, is equal to the previous member put with the same number d (step or the difference of an arithmetic progression). Most often in tasks with arithmetic progressions such questions as finding of the first member of an arithmetic progression, n-go of the member, finding of a difference of an arithmetic progression, the sum of all members of an arithmetic progression are raised. Let's consider each of these questions in more detail.

## It is required to you

- Ability to perform the main mathematical operations.

## Instruction

1. The following communication of the next members of an arithmetic progression - An+1=An+d, for example, A5=6, and d=2, that A6=A5+d=6+2=8 follows from definition of an arithmetic progression.

2. If the first member (A1) and the difference (d) of an arithmetic progression is known, then it is possible to find any her member, I use a formula n-go of the member of an arithmetic progression (An): An=A1+d(n-1). For example, let A1=2, d=5. Let's find, A5 and A10. A5=A1+d(5-1)=2+5(5-1)=2+5*4=2+20=22, and A10=A1+d(10-1)=2+5(10-1)=2+5*9=2+45=47.

3. Using the previous formula it is possible to find the first member of an arithmetic progression. A1 will be then on a formula A1=An-d (n-1) that is if to assume that A6=27, and d=3, A1=27-3(6-1)=27-3*5=27-15=12.

4. To find the difference (step) of an arithmetic progression, it is necessary to know the first and n-y the member of an arithmetic progression, knowing them, the difference of an arithmetic progression is on a formula d=(An-A1) / (n-1). For example, A7=46, A1=4, then d=(46-4)/(7-1)=42/6=7. If d> 0, then the progression is called increasing if d <0 - decreasing.

5. The sum of the first n of members of an arithmetic progression can be found on the following formula. Sn=(A1+An)n/2 where Sn is the sum of n of members of an arithmetic progression, A1, An - the 1st and n-y the member of an arithmetic progression respectively. Let's use data from the previous example, then Sn=(4+46)7/2=50*7/2=350/2=175.

6. If n-y the member of an arithmetic progression is unknown, but the step of an arithmetic progression and number n-go of the member is known, then to find the sum of an arithmetic progression, it is possible to use a formula Sn= (2A1+ (n-1)dn)/2. For example, A1=5, n=15, d=3, then Sn= (2*5+(15-1)*3*15)/2=(10+14*45)/2=(10+630)/2=640/2=320.