In an A.P, if the 7th and 17th terms are 30 and 50 respectively, what is the sum of first 20 terms? The value of the first term = _______?
Answers
Answer :
First term , a = 18
S(20) = 740
Note :
★ A.P. (Arithmetic Progression) : A sequence in which the difference between the consecutive terms are equal is said to be in A.P.
★ If a1 , a2 , a3 , . . . , an are in AP , then
a2 - a1 = a3 - a2 = a4 - a3 = . . .
★ The common difference of an AP is given by ; d = a(n) - a(n-1) .
★ The nth term of an AP is given by ;
a(n) = a + (n - 1)d .
★ If a , b , c are in AP , then 2b = a + c .
★ The sum of nth terms of an AP is given by ; S(n) = (n/2)×[ 2a + (n - 1)d ] .
or S(n) = (n/2)×(a + l) , l is the last term .
Solution :
- Given : a(7) = 30 , a(17) = 50
- To find : First term , a = ? , S(20) = ?
We know that , the nth term of an AP is given by ;
a(n) = a + (n - 1)d .
Thus ,
=> S(7) = a + (7 - 1)d
=> 30 = a + 6d
=> a = 30 - 6d ------(1)
Also ,
=> S(17) = a + (17 - 1)d
=> 50 = a + 16d
=> a = 50 - 16d ------(2)
From eq-(1) and (2) , we have ;
=> 30 - 6d = 50 - 16d
=> 16d - 6d = 50 - 30
=> 10d = 20
=> d = 20/10
=> d = 2
Now ,
Putting d = 2 in eq-(1) , we get ;
=> a = 30 - 6d
=> a = 30 - 6•2
=> a = 30 - 12
=> a = 18
Hence ,
The first term of the AP is 18 .
Also ,
We know that , the sum of first n terms of an AP is given by ;
S(n) = (n/2)•[ 2a + (n - 1)d ]
Thus ,
The sum of first 20 terms of the AP will be given as ;
=> S(20) = (20/2)•[ 2a + (20 - 1)d ]
=> S(20) = 10•[ 2a + 19d ]
=> S(20) = 10•[ 2•18 + 19•2 ]
=> S(20) = 10•[ 36 + 38 ]
=> S(20) = 10•74
=> S(20) = 740