Arithmeticprogression

Arithmetic Progression

Arithmetic Sequence

Arithmetic Sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2. & 2,6,18,54(next term to the term is to be obtained by multiplying by 3.

Arithmetic Sequence

Arithmetic Progression

If various terms of a sequence are formed by adding a fixed number to the previous term or the difference between two successive terms is a fixed number, then the sequence is called AP.

e.g.1) 2, 4, 6, 8, ……… the sequence of even numbers is an example of AP

2) 5, 10, 15, 20, 25….. In this each term is obtained by adding 5 to the preceding term except first term.

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence (an) is given by:

and in general

Arithmetic series

• A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an Arithmetic series.

• The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

Positive, the members (terms) will grow towards positive infinity.

Negative, the members (terms) will grow towards negative infinity.

Common Difference

If we take first term of an AP as a and Common Difference as d. Then--

nth term of that AP will be An = a + (n-1)d. For instance--- 3, 7, 11, 15, 19 … d =4 a =3 Notice in this sequence that if we find the difference

between any term and the term before it we always get 4. 4 is then called the common difference and is denoted

with the letter d. To get to the next term in the sequence we would add 4

so a recursive formula for this sequence is:

The first term in the sequence would be a1 which is sometimes just written as a.

41 nn aa

Example

Let a=2, d=2, n=12,find An

An=a+(n-1)d

=2+(12-1)2 =2+(11)2 =2+22 Therefore, An=24

Hence solved.

The difference between two terms of an AP

The difference between two terms of an AP can be formulated as below:-

nth term – kth term= t(n) – t(k) = {a + (n-1)d} – { a + (k-1) d } = a + nd – d – a – kd + d = nd –

kd Hence, t(n) – t(k) = (n – k) d

General Formulas of AP

• The general forms of an AP is a,(a+d), (a+2d),. .. , a + ( m - 1)d.

i. Nth term of the AP is Tn =a+(n-1)d.

ii. Nth term form the end ={l-(n-1)d}, where l is the last term of the word.

iii. Sum of 1st n term of an AP is Sn=N/2{2a=(n-1)d}.

iv. Also Sn=n/2 (a+1)

v. Tn =(sn-Sn-1)

The sum of n terms, we find as,

Sum = n X [(first term + last term) / 2] Now last term will be = a + (n-1) d

Therefore,

Sum(Sn) =n X [{a + a + (n-1) d } /2 ]

= n/2 [ 2a + (n+1)d]

Sum of n-term of an ap

Problem-- Find number of terms of A.P. 100, 105, 110, 115……500

• Solution.

1) First term is a = 100 , an = 500

2) Common difference is d = 105 -100 = 5

3) nth term is an = a + (n-1)d

4) 500 = 100 + (n-1)5

5) 500 - 100 = 5(n – 1)

6) 400 = 5(n – 1)

7) 5(n – 1) = 400

8) 5(n – 1) = 400

9) n – 1 = 400/5

10) n - 1 = 80

11) n = 80 + 1

12) n = 81

Hence the no. of terms are 81.

MADE BY-: HAFSA KHUNDMIRI

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