Question

the sum of 4th and 8th term of an Ap is 24 and the sum of the 6th and 10 th term is 34 find the 4th term

Answer 1

a4+a8=24. ; a6+a10=34.
a+3d+a+7d=24. ; a+5d+a+9d=34
2a+10d=24......1 ; 2a+14d=35.........2
Solving eq 1,2

2a+10d=24
2a+14d=34
- - -
______________
-4d=-10
d=-10/-4.
d=5/2
putting value of d in eq 1
2a+10×5/2=24
2a+25=24
2a=-1a
a=-1/2
a4=a+3d
=-1/2+3×5/2
=-1 /2+15/2
=-1+15/2
=14/2
a4 =7

Answer 2

${\green {\boxed {\mathtt {✓verified\:answer}}}}$

$let \: a \: be \: the \: first \: term \: and \: d \: be \: the \: common \: difference \: of \: the \: ap \\ then \\ t _{4} + t _{8} = 24 \implies(a + 3d) + (a + 7d) = 24 \\ \implies2a + 10d = 24 \\ \implies \: a + 5d = 12 \: \: \: \: .......(1) \\ and \: t _{6} + t _{10} \implies(a + 5d) + (a + 9d) = 34 \\ \implies2a + 14d = 34 \\ \implies \: a + 7d = 17 ........(2)\\ on \: solving \: (1)and(2) \: we \: get \: a = -1/2 \: and \: d= 5/2 \\ \therefore \: fourth \: terms \: is \: 7$