# Arithmetic Progressions

If an auditorium has 10 rows and each row has 15 seats, then there are (10) (15) = 150 seats.But a typical auditorium is narrower at the front than at the back (see in figure given below).

Suppose that successive rows lengthen by a fixed number of seats. Figure shows that each row (other than the front now) has 2 seats more than the row in front of it.

The front row has 10 seats, the next 12 seats, and next 14 seats and so on. What is the number of seats in the 10th row? If the auditorium has 50 rows, what is the number of seats in the 50th row?

In case of 10 rows, we may write the sequence

10, 12, 14, 16, 18, 20, 22, 24, 26, 28

Thus, the number of seats in the 10th row is 28.

In case of 50 rows, the above method will involve a lot of labour. Let us try to develop a shortcut.

Consider an auditorium in which the front row has *a* seats. The next row has *d* more seats, thus the second row has

*a* + *d* seats the third row had *d *more seats than the second row, hence the third row has

*a* + 2*d* seats.

the fourth row

*a* + 3*d* seats.

and so on. If there are n rows in the auditorium, the *n ^{th}* row has

*a* + (*n* â€“ 1) *d* seats.

The sequence

a, *a* + *d*, *a* + 2* d*, *a* + 3* d*, â€¦, *a* + (*n* â€“ 1) *d*, â€¦ is called an arithmetic progression.

Let us take a few more examples, before going to formal definition.

Consider the sequence 1, 4, 7, 10, 13, 16, â€¦ (1)

or the sequence 25, 15, 5, âˆ’ 5, âˆ’ 15 â€¦ (2)

In (1), we note that *t*_{1} = 1, *t*_{2} = 4 = *t*_{1} + 3, *t*_{3} = 7 = *t*_{2}* *+ 3, *t*_{4} = 10 = *t*_{3 }+ 3 and so on.

We find that each term except the first â€˜*progresses*â€™ in a definite manner.

In (2), we note that *t*_{1} = 25, *t*_{2} = 15 = *t*_{1} â€“ 10, *t*_{3} = 5 = *t*_{2} â€“ 10, *t*_{4} =âˆ’ 5 = *t*_{3} â€“ 10 and so on.

We once again find that each term except the first â€˜progressâ€™ in a definite manner.

We observe that each term except the first is obtained by adding a fixed constant to the term immediately preceding it. Such sequences are called arithmetic sequences or arithmetic progressions or briefly AP.

Definition

An arithmetic sequence or arithmetic progression is a sequence

*a*

_{1, }a

_{2}, a

_{3}, a

_{4}, â€¦

which is such that a

_{2}â€“ a

_{1}= a

_{3}â€“ a

_{2}= a

_{4}â€“ a

_{3}= â€¦

That is, in an arithmetic progression each pair of consecutive terms differs by the same constant.

If *d*** **= a_{2} â€“ a_{1} = a_{3} â€“ a_{2} = *a _{4}*â€“ a

_{3}= â€¦ then

*d*

**is called as the common difference of the arithmetic progression.**

Note that we can write

*a*

_{i + 1}=

*a*+

_{i}*d*for each

*i*ÃŽ N. This is called a recursive formula, because it defines a given term by reference to the preceding term.

Illustrations

**Some examples of arithmetic progressions are as follows:**

(i) 9, 15, 21, 27, 33, â€¦

Thus common difference in this case is 6 and

*a*

_{n +1}=

*a*+ 6 âˆ€

_{n}*n*ÃŽ N.

(ii) Ï€ + 1, Ï€ + 2, Ï€ + 3, â€¦

In this case the common difference is 1 and

*a*

_{n}_{ +1}=

*a*+ 1âˆ€

_{n}*n*ÃŽ N.

(iii) â€“1, , , , â€¦

In this case the common difference is âˆ’ (âˆ’ 1) = and

a _{n}_{ +1} = *a _{n}* +âˆ€

*n*ÃŽ N.

(iv) *x* + *y*, *x* â€“ *y*, *x* â€“ 3 *y*, *x* â€“ 5 *y*, â€¦

In this case the common difference is (*x* â€“ *y*) â€“ (*x*s + *y*) = - 2 *y* and

a _{n + 1} = *a* _{n} - 2 *y* âˆ€ *n* ÃŽ N.

If we know the first term *a*_{1} of an arithmetic progression and the common difference d, we can form the other terms by repeatedly adding *d*.

a_{1} = a_{1} = a_{1} + (0) *d *= a_{1} + (1 â€“ 1) *d*

a_{2} = a_{1} + *d *= a_{1} + (1) *d* = a_{1} + (2 â€“ 1)* d*

a_{3} = a_{1} + *d* + *d* = a_{1} + (2) *d* = a_{1} + (3 â€“ 1)* d*

a_{4} = a_{1} + *d* + *d* + *d* = a_{1} + (3) *d* = a_{1} + (4 â€“ 1)* d*

a_{5} = a_{1} + *d* + *d* + *d* + *d* = a_{1} + (4) *d* = a_{1} + (5 â€“ 1) *d* and so on.