In an A.P., the sum of the first three terms is 12 and the sum of the next three tersms is 30. Find the first term
Answers
Step-by-step explanation:
Given:-
In an A.P., the sum of the first three terms is 12 and the sum of the next three terms is 30.
To find:-
Find the first term ?
Solution:-
Let the first term of an AP = t1
Common difference = d
nth term of an AP = tn = t1 + (n-1)d
Given that
The sum of first three terms in an AP = 12
=> t1 + t2 + t3 = 12
=> t1 + t1 + d + t1 + 2d = 12
=> 3 t1 + 3d = 12
=> 3( t1 + d ) = 12
=> t1 + d = 12/3
=> t1 + d = 4---------------(1)
and
Sum of next three terms = 30
t4 + t5 + t6 = 30
=> t1 + 3d + t1 + 4d + t1 + 5d = 30
=> 3t1 + 12d = 30
=> 3( t1 + 4d ) = 30
=> t1 + 4d = 30/3
=> t1 + 4d = 10 -----------(2)
On subtracting (1) from (2)
t1 + 4d = 10
t1 + d = 4
(-)
__________
0 + 3d = 6
__________
=> 3d = 6
=> d = 6/3
=> d = 2
On substituting this value in (1) then
=> t1 + 2 = 4
=> t1 = 4-2
=>t1 = 2
We have t1 = 2 and d = 2
First term = 2
Answer:-
The first term of the given AP = 2
Used formula:-
- nth term of an AP = tn = t1 + (n-1)d
- t1 = First term
- d = Common difference
- n = number of terms