Class 10 Maths Arithmetic Progressions nth term of AP

nth term of Arithmetic Progression

Let a1, a2, a3, . . . be an AP whose first term a1 is a and the common difference is d., thus the AP is Thus a, a + d, a + 2d, a + 3d,

The first term a1 = a + 0*d = a + (1 – 1) d

The second term a2 = a + d = a + (2 – 1) d

The third term a3 = a + 2d = a + (3 – 1) d

The fourth term a4 = a + 3d = a + (4 – 1) d

. . . . . . . .

Looking at the pattern, we can say that the nth term an = a + (n – 1) d.

So, the nth term an of the AP with first term ‘a’ and common difference d is given by an = a + (n – 1) d.

an is also called the general term of the AP

Example 1: Find the 10th term of the AP : 2, 7, 12, . . .

Solution: Here, a = 2, d = 7 – 2 = 5 and n = 10.

We have an = a + (n – 1) d

So, a10 = 2 + (10 – 1) × 5 = 2 + 45 = 47

Therefore, the 10th term of the given AP is 47.

Example 2: Determine the AP whose 3rd term is 5 and the 7th term is 9.

Solution: a3 = a + (3 – 1) d = a + 2d = 5    --(i)

a7 = a + (7 – 1) d = a + 6d = 9                       --(ii)

Solving the pair of linear equations (i) and (ii), we get a = 3, d = 1

Hence, the required AP is 3, 4, 5, 6, 7 . . .

Example 3 : Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .

Solution:  d= a2a­ = 6 & a=5

Let 301 be a term, say, the nth term of this AP.

an = a + (n – 1) d

or 301 = 5 + (n – 1) × 6

301 = 6n – 1

Or   n = 302/6 which is a fraction. Thus 301 is not in the series.

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