Question

find value of a,b and c such that a,10,b,c,31 are in AP

Answer 1

a,10,b,c,31 are in AP

c
let the common difference = d

second term = 10
a+d=10----(1)
fifth term = 31
a+4d=31---(2)
subtract (1) from (2)
3d=21
⇒d=21/3
d=7
put d=7 in (1)
a+7=10
⇒a=10-7
a=3
therefore
a=3 , d=7
third term=b = a+2d = 3+2*7 = 3+14 =17
fourth term = c= a+3d = 3+3*7 =3+21=24
a=3, b=17,c=24

Answer 2

a,10,b,c,31 -----> are in A.P

By Arithmatic Mean ;
If a,b,c are in A.P then , 2b = a+b
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Similarly ;

20 = a+b -------(1)

2b = 10+c -------(2)

2c = 31+b -------(3)
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c = (31/2)+(b/2)

2b = 10+(31/2)+(b/2)
2b-(b/2) = (51/2)
3b = 51
b = 17

20 = a+b
20 = a+17
a = 3

2c = 31+b
2c = 31+17
2c = 48
c = 24
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Hence the value of a,b & c are 3,17,24 respectively.