An A.P. consists of 41 terms. The sum of the middle three terms is 237 and the sum of
the last three terms is 465. Find the A.P.
Answers
Step-by-step explanation:
Given :-
An A.P. consists of 41 terms. The sum of the middle three terms is 237 and the sum of
the last three terms is 465.
To find :-
Find the A.P.?
Solution :-
Given that
The number of terms of an AP (n) = 41
The middle term =(n+1)/2
=> (41+1)/2
=> 42/2
=> 21
So, the middle three terms are 20,21 and 22 terms
We know that
nth term of an AP (an) = a+(n-1)d
20th term = a+(20-1)d = a+19d
21st term = a+(21-1)d = a+20d
22nd term = a+(22-1)d = a+21d
Given that
The sum of the middle three terms = 237
=> a20 + a21 + a22 = 237
=> (a+19d)+(a+20d)+(a+21d) = 237
=> (a+a+a) +(19d+20d+21d) = 237
=> 3a+60d = 237
=> 3(a+20d) = 237
=> a+20d = 237/3
=> a+20d = 79 ---------------(1)
and
The last three terms are 39 , 40 and 41 terms
Given that
The sum of the last three terms = 465.
=> a39 + a40 + a41 = 465
=> a+38d + a+39d + a+40d = 465
=> (a+a+a)+(38d+39d+40d) = 465
=> 3a + 117d = 465
=> 3(a+39d) = 465
=> a+39d = 465/3
=> a+39d = 155 -------------(2)
On subtracting (1) from (2) then
a+39d = 155
a+20d = 79
(-) (-) (-)
___________
0+19d = 76
___________
=> 19d = 76
=> d = 76/19
=> d = 4
Therefore, Common difference = 4
On substituting the value of d in (1) then
=> a +20(4) = 79
=> a+80 = 79
=> a = 79-80
=> a = -1
Therefore, First term = -1
Now
The general form of an AP = a,a+d,a+2d,....
a = -1
a+d = -1+4 = 3
a+2d = -1+2(4) = -1+8 = 7
a+40d = -1+40(4) = -1+160 = 159
The AP : -1 , 3 , 7 ,..., 159
Answer:-
The required AP for the given problem is
-1, 3 , 7, ..., 159
Used formulae:-
→ The general form of an AP = a,a+d,a+2d,....
→ nth term of an AP (an) = a+(n-1)d
→ a = First term
→ d = Common difference
→ n = number of terms