Those sequence whose terms follow certain patterns are called progression. Generally there are three types of progression.
(i) Arithmetic Progression (A.P.)
(ii) Geometric Progression (G.P.)
(iii) Harmonic Progression (H.P.)
1.2 ARTHMETIC PROGRESSION :
A sequence is called an A.P., if the difference of a term and the previous term is always same. i.e.
d = tn+1 – tn
= Constant for all
nN
. The constant difference, generally denoted by ‘d’ is called the common difference.
Ex.1 Find the common difference of the following A.P. : 1,4,7,10,13,16 ......
Sol. 4 - 1 = 7 - 4 = 10 - 7 = 13 - 10 = 16 - 13 = 3 (constant).
Common difference (d) = 3.
6.3 GENERAL FORM OF AN A.P. :
If we denote the starting number i.e. the 1st number by ‘a’ and a fixed number to the added is ‘d’ then a, a +
d, a + 2d, a + 3d, a + 4d, ...... forms an A.P.
Ex.2 Find the A.P. whose 1st term is 10 & common difference is 5.
Sol. Given : First term (a) = 10 & Common difference (d) = 5.
A.P. is 10, 15, 20, 25, 30, .....
6.4 nth TERM OF AN A.P. :
Let A.P. be a, a + d, a + 2d, a + 3d, .....
Then, First term (a1
) = a + 0.d
Second term (a2
) = a + 1.d
Third term (a3
) = a + 2.d
. .
. .
. .
nth term (an) = a + (n - 1) d
an = a + (n - 1) d is called the nth term.
Ex.3 Determine the A.P. whose their term is 16 and the difference of 5th term from 7th term is 12.
Sol. Given : a3
= a + (3 - 1) d = a + 2d = 16 .....(i)
a7
- a5
= 12 ....(ii)
(a + 6d) - (a + 4d) = 12
a + 6d - a - 4d = 12
2d = 12
d = 6
Put d = 6 in equation (i)
a = 16 - 12
a = 4
A.P. is 4, 10, 16, 22, 28, ......
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