# 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 ?

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

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

And we know nth term in A.P.  :

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

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

a + 4d   x                               ------- ( 1 )
And
8th term
T8  = a  + ( 8 -  1 ) d   =   a + 7d , Gven : 8th 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 ) 6th term
T6  =  -7x + ( 6 -  1 ) 2x

T6  =  -7x +10x

T6  = 3x
And

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

T15  =  -7x +28x

T15  = 21x
So,

So,
The ratio between 6th and 15th terms   =  1 : 7                                                   ( Ans )

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

T7  =  -7x +12x

T15  = 5x  , and given 7th 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,
22nd term of this sequence  = -14 + ( 22 - 1 ) 4  =   -14  + ( 21 )4 =  -14  + 84  =  70                                     ( Ans )

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