The sum of 4th and 8th terms of an A.P. is 24, and the sum of 6th and 10th terms is 44. Find the A.P.
Answers
Step-by-step explanation:
Given:-
The sum of 4th and 8th terms of an A.P. is 24, and the sum of 6th and 10th terms is 44.
To find :-
Find the A.P ?
Solution:-
First term is 'a' and the common difference is 'd' and "n" is the number of terms of an AP then the general term is an = a+(n-1)d
4th term of an AP = a4= a+(4-1)d= a+3d
8th term of the AP = a8 = a+(8-1)d = a+7d
Given that
The sum of 4th and 8th terms of an AP =24
=>(a+3d) + (a+7d) = 24
=>a+3d+a+7d = 24
=>2a + 10d = 24
=>2(a+5d) = 24
=>(a+5d) = 24/2
a+5d = 12 -----------(1)
6th term = a+(6-1)d = a+5d
10th term = a+(10-1)d = a+9d
the sum of 6th and 10th terms is 44
=>(a+5d)+(a+9d) = 44
=>a+5d+a+9d = 44
=>2a +14d = 44
=>2(a+7d) = 44
=>a+7d = 44/2
a+7d = 22 ------------(2)
on solving (2)&(1)
a+7d = 22
a+5d = 12
(-)
_________
0+2d = 10
________
=>2d = 10
=>d =10/2
=>d = 5
Common difference of the AP = 5
from (1) we have
a+5(5)= 12
=>a+25 = 12
=>a = 12-25
=>a = -13
First term of the AP = -13
now The general form of the AP is
a,a+d,a+2d,....
a = -13
a+d = -13+5 = -8
a+2d = -13+2(5) = -13+10= -3
The AP = -13 , -8 , -3 ,...
Answer:-
The Arithmetic Progression for the given problem is -13 , -8 , -3 , 2 , 7 , 12 , ....
Check:-
a4 = a+3d
=>a4 = -13+3(5)
=>a4 = -13+15
a4 = 2
a8 = a + 7d
=>a 8 = -13+7(5)
=>a8 = -13+35
a8 = 22
a4 + a8
=>2+22
=>24
and
a6 = a +5d
=>a6 = -13+5(5)
=>a6 = -13+25
a6 = 12
a10 = a +9d
=>a 10 = -13+9(5)
=>a10 = -13+45
=>a10 = 32
a6 +a10
=>12+ 32
=>44
verified the given relations
☆ Correct Question ☆
The sum of 4th and 8th terms of an A.P. is 24, and the sum of 6th and 10th terms is 44. Find the A.P.
☆Step by step explanation☆
Given:-
> The sum of 4th and 8th terms of an A.P. is 24, and the sum of 6th and 10th terms is 44.
To find :-
what is the A.P ?
Approach :-
First term is 'a' and the common difference is 'd' and "n" is the number of terms of an AP then the general term is an = a+(n-1)d
Solution :-
4th term of an AP = a4= a+(4-1)d= a+3d
8th term of the AP = a8 = a+(8-1)d = a+7d
The sum of 4th and 8th terms of an AP =24
=>(a+3d) + (a+7d) = 24
=>a+3d+a+7d = 24
=>2a + 10d = 24
=>2(a+5d) = 24
=>(a+5d) = 24/2
a+5d = 12 -----------(1)
>6th term = a+(6-1)d = a+5d
>10th term = a+(10-1)d = a+9d
the sum of 6th and 10th terms is 44
=>(a+5d)+(a+9d) = 44
=>a+5d+a+9d = 44
=>2a +14d = 44
=>2(a+7d) = 44
=>a+7d = 44/2
a+7d = 22 ------------(2)
on solving (2)&(1)
a+7d - (a+5d) = 22 - 12
0+2d = 10
=>2d = 10
=>d =10/2
=>d = 5
•°• Common difference of the AP = 5
from (1) we have
a+5(5)= 12
=>a+25 = 12
=>a = 12-25
=>a = -13
⚽ATQ⚽
First term of the AP = -13
now,
The general form of the AP :-
a,a+d,a+2d,....
a = -13
a+d = -13+5 = -8
a+2d = -13+2(5) = -13+10= -3
So, AP = -13 , -8 , -3 ,...
Hence, Arithmetic Progression is -13 , -8 , -3 , 2 , 7 , 12 , ....
⚽ Verification ⚽
a4 = a+3d
=>a4 = -13+3(5)
=>a4 = -13+15
•°• a4 = 2
a8 = a + 7d
=>a 8 = -13+7(5)
=>a8 = -13+35
•°• a8 = 22
Now,
a4 + a8
=>2+22
=>24
⚽Hence, verified ⚽
#similar Question to practice:-
> Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0?
>Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .
>If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively, then which term of this A.P is zero.
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