The 5th and 8th term of an a.p is in the ratio 1:7 .
(a) what is the ratio between 6th and 15th terms ?
(b) The 7th term of this sequence is 10 . What is 22nd term ?

Answer :

Given  : The 5^th and 8^th term of an a.p is in the ratio 1:7
Let ratio coefficient  =  x , So

5^th term of A.P. =  x
And
8^th term of A.P.  = 7x

And we know n^th term in A.P.  :

T_n =  a + ( n  -  1 ) d   , Here a  =  first term and d  =  Common difference

So, 5^th term
T₅  = a  + ( 5 -  1 ) d   =   a + 4d , Gven : 5^th term of A.P. =  x So

  a + 4d   =  x                               ------- ( 1 )
And
8^th term
T₈  = a  + ( 8 -  1 ) d   =   a + 7d , Gven : 8^th term of A.P. =  7x So

  a + 7d   =  7x                               ------- ( 2 )
Now we subtract equation 1 from equation 2 , we get

3d  = 6x

d  = 2x               , Substitute in equation 1 , we get

a  + 4 ( 2x ) = x

a  = x  - 8x

a  = - 7x
So,
a ) 6^th term
T₆  =  -7x + ( 6 -  1 ) 2x  

T₆  =  -7x +10x  

T₆  = 3x  
And

15^th term
T₁₅  =  -7x + ( 15 -  1 ) 2x  

T₁₅  =  -7x +28x  

T₁₅  = 21x  
So,
$\frac{6^{th} term }{{15}^{th} term} = \frac{T_6}{T_{15}} = \frac{3x}{21x} = \frac{1}{7}$
So,
The ratio between 6^th and 15^th terms   =  1 : 7                                                   ( Ans )

b ) 7^th term
T₇  =  -7x + ( 7-  1 ) 2x  

T₇  =  -7x +12x  

T₁₅  = 5x  , and given 7^th term of this sequence  =  10 , So

5x =  10 

x  =  2
So,
First term a  = -7x  =  -7 ( 2 ) =  -14
And
Common difference = d  = 2x  = 2 ( 2 ) = 4
So,
22^nd term of this sequence  = -14 + ( 22 - 1 ) 4  =   -14  + ( 21 )4 =  -14  + 84  =  70                                     ( Ans )