Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4th term is :

  • Option 1)

    8

  • Option 2)

    16

  • Option 3)

    20

  • Option 4)

    24

 

Answers (2)

As we learnt in 

Sum of n terms of an AP -

S_{n}= \frac{n}{2}\left [ 2a +\left ( n-1 \right )d\right ]

and

Sum of n terms of an AP

S_{n}= \frac{n}{2}\left [ a+l\right ]

- wherein

a\rightarrow first term

d\rightarrow common difference

n\rightarrow number of terms

given 200 < S< 220

and a+d=12 -----(i)

Then a+3d=?

\therefore S_{9}=\frac{9}{2}\left [ 2\cdot a+(9-1)d \right ]

         =\frac{9}{2}[ 2a+8d]

         =\frac{9}{2}[ 2a+8(12-9)]

         =\frac{9}{2}[ 96-6a]

          =\frac{9}{2}[48-3a]

Now    200 < 9(48 - 3a) <220

          \therefore a=8 (because a is positive integer)

          \therefore d=4

\therefore a+3d=8+12=20

        


Option 1)

8

This option is incorrect

Option 2)

16

This option is incorrect

Option 3)

20

This option is correct

Option 4)

24

This option is incorrect

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