Question

If \[\frac{5 + 9 + 13 + . . . \text{ to n terms} }{7 + 9 + 11 + . . . \text{ to (n + 1) terms}} = \frac{17}{16},\] then n =

 

Options

• 8

• 7

• 10

• 11

Answer 1

Here, we are given,

\[\frac{5 + 9 + 13 + . . . \text{ to n terms} }{7 + 9 + 11 + . . . \text{ to (n + 1) terms}} = \frac{17}{16},\]             ...........(1) 

We need to find n.

So, first let us find out the sum of n terms of the A.P. given in the numerator ( 5 + 9 + 13 + ...) . Here we use the following formula for the sum of n terms of an A.P.,

`S_n = n/2 [ 2a + ( n - 1) d ]`

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

Here,

Common difference of the A.P. (d) =  a₂ - a₁ 

_{= 9 - 5}

_{= 4}

 Number of terms (n) = n

First term for the given A.P. (a) = 5 

So, using the formula we get,

`S_n = n/2 [ 2(5) + ( n-1) ( 4) ] `

     `= (n/2) [ 10 + (4n - 4)] `

     ` = ( n/2) (6 +4n) ` 

      = n (3 + 2n)                ....(2) 

Similarly, we find out the sum of ( n + 1)  terms of the A.P. given in the denominator ( 7 + 9 + 11+...) .

Here,

Common difference of the A.P. (d) = a₂- a₁ 

= -9 - 7

= 2 

Number of terms (n) = n

First term for the given A.P. (a) = 7 

So, using the formula we get,

`S_(N+1) = ( n+1)/2 [ 2(7) + [( n + 1 ) - 1](2)]`

        `= ((n + 1) /2 )[14 + ( n) ( 2 ) ] `

        = ( n + 1 ) ( 7 + n) 

        = 7n + 7 + n² + n 

        = n² + 8n + 7              ..........(3) 

Now substituting the values of (2) and (3) in equation (1), we get,

     `(2n^2 + 3n ) /(n^2 + 8n + 7) = 17/16`

                         16 (2n² + 3n ) = 17 ( n²+ 8n + 7)

                              32n² + 48n = 17n²+ 136n + 119

32n² - 17n² + 48n - 136n - 119 = 0

                       15n² - 88n - 119 = 0

 

Further solving the quadratic equation for n by splitting the middle term, we get,

             15n² - 88n - 119 = 0

15n² - 105n + 17n +- 119 = 0

    15n ( n - 7) + 17 ( n - 7) = 0

               ( 15n + 17 )(n- 7) = 0

So, we get

15n + 17 = 0

       15n = - 17

          `n = (-17)/15`

Or

n - 7 = 0

     n = 7 

Since n is a whole number, it cannot be a fraction. So, n = 7