Find the respective terms for the following APs. (i) a2=38,a6=-22 then find a1, a3, a4, a5
Answers
Given : In an AP, a2 = 38 and a6 = - 22
To find : a1, a3, a4 and a5
Solution :
AP ( Arithmetic Progression ) is an arithmetic sequence of numbers in which the common difference between two consecutive terms is always same.
To solve the given problem, firstly you should have some basic knowledge of nth term.
nth term of an AP is represented by an which can be calculated by following formula :
- an = a + ( n - 1 ) d
Here,
- a = First term
- n = Number of terms
- d = Common difference
- an = nth term
Given that the 2nd term ( a2 ) is 38. It can be expressed as,
=> a2 = a + (2 - 1) d
=> 38 = a + d - - - - Eqn(1)
It is also given that the 6th term ( a6 ) is - 22, It can be expressed as,
=> a6 = a + ( 6 - 1 ) d
=> -22 = a + 5d - - - - Eqn(2)
Now try to examine Eqn(1) and Eqn(2), we can eliminate variable a to get the value of d.
Subtract Eqn(2) from Eqn(1)
=> -22 - 38 = a + 5d - ( a + d )
=> -60 = a + 5d - a - d
=> -60 = 4d
=> -60 / 4 = d
=> -15 = d
So the common difference of AP is -15.
Put this value of d in Eqn(1)
=> 38 = a + d
=> 38 = a - 15
=> 38 + 15 = a
=> 53 = a
So the first term of AP is 53.
Since we have obtained the value of first term and common difference, it's quite easy to find any term of given AP.
(i)
=> a1 = a = 53
(ii)
=> a3 = a + (3 - 1) d
=> a3 = a + 2d
=> a3 = 53 + 2(-15)
=> a3 = 53 - 30
=> a3 = 23
(iii)
=> a4 = a + (4 - 1)d
=> a4 = a + 3d
=> a4 = 53 + ( 3 ) ( - 15 )
=> a4 = 53 - 45
=> a4 = 8
(iv)
=> a5 = a + (5 - 1) d
=> a5 = a + 4d
=> a5 = 53 + 4 ( -15 )
=> a5 = 53 - 60
=> a5 = -7
So the required answers are :-
- a1 = 53
- a3 = 23
- a4 = 8
- a5 = -7
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