Question

In HP t4 =1/11 and t14=3/23 find t7

Answer 1

t4=1/11
t4=a+3d
1/11=a+3d
1=11(a+3d)
1=11a+33d
11a=1-33d
a=1-33d/11 ........(i)

Again,
t14=3/23
a+13d=3/23
23(a+13d)=3
23a+299d=3
23a=3-299d
a=3-299d/23 .........(ii)

From (i) & (ii),

1-33d/11=3-299d/23
23(1-33d)=11(3-299d)
23-759d=33-3289d
3289d-759d=33-23
2530d=10
d=10/2530
d=1/253..........(iii)

Put (iii) in (i),
a=1-33d/11
a=1-33×(1/253)÷11
a=1-33/253÷11
a=220/253÷11
a=20/253

Now,
t7=a+6d
=20/253+6×1/253
=20/253+6/253
=26/253

Therefore,,t7=26/253