Arithmatic progression

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05 ARITHMATIC PROGRESSION

Arithmetic Progression:

An arithmetic progression is list of numbers in which the difference between any two consecutive terms is constant.

a_{1,} a_{2}, a_{3}, a_{4} . . . . . . . . a r e i n A . P . i f

a_{2} - a_{1} = d = a_{4} - a_{3} = a_{5} - a_{4} . . . . . d = c o m m o n d i f f e r e n c e a = a_{1} F i r s t t e r m .

e.g 4,710,13,…… is A. P with a=4 and d=3

Each member of the sequence is called term.

Common difference d can be zero, positive or negative.
General form of AP is: a,a+d,a+2d
A.P. can be finite or infinite.

General Term of A.P. THEOREM Let ‘a ’ be the first term and ‘d ’ be the common difference

o f a n A . P . T h e n, i t s n^{t h} t e r m o r g e n e r a l t e r m i s g i v e n b y a_{n} = a + (n - 1) d

PROOF

L e t a_{1}, a_{2}, . . . . . a_{n} b e t h e g i v e n A . P . T h e n a_{1} = a,

\to a_{1} = a + (1 - 1) d . . . . . . (1)

Since each term of an A.P. is obtained by adding common difference to the preceding term. Therefore,

a_{2} = a + d

\to a_{2} = a + (2 - 1) d

………(2)

Similarly, we have

a_{3} = a_{2} + d

\to a_{3} = (a_{} + d) + d

\to a_{3} = a + 2 d

\to a_{3} = a + (3 - 1) d

\to a_{4} = (a + 2 d) + d . . . . . . (3)

\to a_{4} = a + 3 d

a_{4} = a + (4 - 1) d . . . . (4)

Observing the pattern in equation (1), 2), (3) and (4), we find that

a_{n} = a + (n - 1) d

n^{t h} T E R M O F A N A . P . F R O M T H E E N D

Let there be an A.P. with first term ‘a ’ and common difference ‘d ’. If there are ‘m ’ terms in

t h e A . P ., t h e n n^{t h} t e r m f r o m t h e e n d = (m n + 1)^{t h} t e r m f r o m t h e b e g i n n i n g

\to n^{t h} t e r m f r o m t h e e n d = a_{m - n + 1}

\to n^{t h} t e r m f r o m t h e e n d

= a + (m - n + 1 - 1) d

n^{t h} t e r m f r o m t h e e n d = a + (m - n) d

Also, if I is the last term of the A.P., then term from the end is the n^th term of an A.P. whose first term is I and common difference is –d.

∴ n^{t h} t e r m f r o m t h e e n d = L a s t t e r m + (n - 1) (- d)

n^{t h} t e r m f r o m t h e e n d = I - (n - 1) d

MIDDLE TERM(S) OF A FINITE A.P.

Let there be a finite A.P. with ‘ ’n terms whose first term is ‘a ’ and common difference is ‘d’.

If n is odd, then

{(\frac{n + 1}{2})}^{t h}

term is the middle term and is given by

a + (\frac{n + 1}{2} - 1) d

If n is even, then

{(\frac{n}{2})}^{t h} a n d {(\frac{n}{2} + 1)}^{t h}

are middle term given by

a + (\frac{n}{2} - 1) d

a n d a + (\frac{n}{2} + 1 - 1) d = a + \frac{n}{2} d

respectively

Sum of n terms of AP:

Sum of first n terms of A.P. is given by

s_{n} = \frac{n}{2} [2 a + (n - 1) d]

o r s_{n} = \frac{n}{2} [a + a_{n}]

o r_{s_{n} = \frac{n}{2} [a + 1]} (1 = l a s t t e r m o f A . P .)

• T h e n^{t h}

• t e r m a l s o c a n b e g i v e n b y a_{n} = s_{n} - s_{n - 1}

THEOREM The sum of n terms of an A.P. with first term ‘ a’ and common difference ‘ d’ is

s_{n} = \frac{n}{2} \{2 a + (n - 1) d\}

o r s_{n} = \frac{n}{2} \{a + a + (n - 1) d\}

o r s_{n} = \frac{n}{2} \{a + 1\}

, w h e r e 1 = l a s t t e r m = a + (n - 1) d

P R O O F : L e t a_{1,} a_{2}, a_{3} b e a n A . P . w i t h f i r s t t e r m ‘ a ’ a n d c o m m o n d i f f e r e n c e ‘ d ’ . T h e n,

a_{1} = a, a_{2} = a + d, a_{3} = a + 2 d, a_{4} = a + 3 d, . . . . . . . ., a_{n} = a + (n - 1) d

N o w, s_{n} = a_{1} + a_{2} + a_{3} + . . . . . . . . + a_{n}

\to s_{n} = a + (a + d) + (a + 2 d) + . . . . . . . + (a + (n - 2) d) + {a + (n - 1) d} . . . . . . (1)

Writing the above series in a reverse order, we get

s_{n} = {a + (n - 1) d} + {a + (n - 2) d} + . . . . . . . + (a + d) + a . . . . . . . . . (2)

Adding the corresponding terms of equations (1) and (2), we get

2 s_{n} = {2 a + (n - 1) d} + {2 a + (n - 1) d} + . . . . . . + {2 a + (n - 1) d}

2 s_{n} = n {2 a + (n - 1) d}

[∵ 2 a + (n - 1) d r e p e a t s n t i m e s]

\to s_{n} = \frac{n}{2} {2 a + (n - 1) d}

Now,

I = l a s t t e r m = n^{T H} t e r m = a + (n - 1) d

∴ s_{n} = \frac{n}{2} {2 a + (n - 1) d} = \frac{n}{2} [a + {a + (n - 1) d}] = \frac{n}{2} {a + 1}

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