The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
Answers
Answer:
The first term is - ½ and common difference is 5/2.
Step-by-step explanation:
Given:
a4 + a8 = 24 …………..(1)
a6 + a10 = 34 ………….(2)
Formula used : an = a + (n - 1)d
a4 + a8 = 24
a +(4 -1)d + a +(8 - 1) = 24
a + 3d + a + 7d = 24
2a + 10d = 24
2(a + 5d) = 24
a + 5d = 24/2
a + 5d = 12 ……. ……(3)
a6 + a10 = 34
a + (6 - 1)d + a + (10 -1)d = 34
a + 5d + a + 9d = 34
2a + 14d = 34
2(a + 7d) = 34
a + 7d = 34/2
a + 7d = 17…………….(4)
On Subtracting eq (3) from (4)
(a + 7d) - (a + 5d) = 17 - 12
a + 7d - a - 5d = 17 - 12
2d = 5
d = 5/2
common difference ,d = 5/2
On Putting the value of d = 5/2 in (3),
a + 5d = 12
a + 5(5/2) = 12
a + 25/2 = 12
a = 12 - 25/2
a = (24 - 25)/2
a = - ½
First term , a = - ½
Hence, the first term is - ½ and common difference is 5/2.
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SOLUTION☺️
=)let the first term of an AP= a
and the common difference of the given A.P. = d
AS WE KNOW THAT,
=) a n = a+(n-1)d
=) a4= a+(4-1)d
=)a4= a+3d
similarity,
=)a8= a+7d
=)a6= a+5d
=)a10= a+9d
Sum of 4th and 8th term = 24(GIVEN)
=)a4+a8= 24
=) a+3d+a+7d=24
=)2a+10d= 24
=) a+5d= 12-----------------(1)
Sum of 6th and 10th term= 34(GIVEN)
=) a6+a10= 34
=)a+5d+a+9d= 34
=)2a+14d= 34
=) a+7d= 17-----------------(2)
Solving (1)&(2), we get
=) a+7d= 17
a+5d= 12
- - -
_____________'
2d= 5
=) d= 5/2
From equation (1), we get
=) a+5d= 12
=) a+5(5/2)= 12
=)a+25/2= 12
=)2a+25=24
=)2a= 24-25
=)2a= -1
=)a= -1/2
a2= a+d= -1/2+5= 9/2
=) a3= a2+d= 9/2+5= 19/2
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