If sum of two terms is 10 and sum of first five terms is 20 the sum of three term is
Answers
Question :
In an AP , if the sum of first two terms is 10 and sum of first five terms is 20 then what is the sum of three terms ?
Answer :
S(3) = 14
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) = a1 + (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 ] .
★ The nth term of an AP can be also given by ; a(n) = S(n) - S(n-1) .
★ A linear polynomial in variable n always represents the nth term of an AP .
★ A quadratic polynomial in variable n always represents the sum of n terms of an AP .
★ If each terms of an AP is multiplied or divided by same quantity , then the resulting sequence is an AP .
★ If same quantity is added or subtracted in each term of an AP then the resulting sequence is an AP .
Solution :
- Given : S(2) = 10 , S(5) = 20
- To find : S(3) = ?
Let the five terms of the AP as ;
(a - 2d) , (a - d) , a , (a + d) , (a + 2d)
Now ,
According to the question ,
=> S(2) = 10
=> (a - 2d) + (a - d) = 10
=> 2a - 3d = 10 --------(1)
Also ,
=> S(5) = 20
=> (a-2d) + (a-d) + a + (a+d) + (a+2d) = 20
=> 5a = 20
=> a = 20/5
=> a = 4
Now ,
Sum of first three terms wil be ;
=> S(3) = (a - 2d) + (a - d) + a
=> S(3) = (2a - 3d) + a
=> S(3) = 10 + 4 {using eq-(1)}
=> S(3) = 14