Question

The sum of first six term of an ap is 42 the ratio of ten term of its 30 term is 1:3 find the first ap

Answer 1

given,
  
           $S_{6}$ = 42

           $\frac{a_{10} }{ a_{30} } = \frac{1}{3}$

           $S_{6} = a + 5d$

           42 = a +5d..............(1)

           $\frac{a+9d}{a+29d} = \frac{1}{3}$

           3( a +9d ) = a +29d

           3a + 27d = a + 29d

            2a = 2d

             a = d .........................(2)

now by substituting (2) equation in (1) we will get the first term of the A.P

    42 = a + 5d

   42 = a +5a

   42 = 6a

     a= 7