Question

The product of the third by the sixth term of an arithmetic progression is $ 406 $ . The division of the ninth term of the progression by the fourth term gives a quotient $ 2 $ and a remainder $ 6 $ . Find the first term and the difference of the progression.

Answer 1

Hint: Here we will convert the given word statements in the form of mathematical expression by using the standard equation for the arithmetic expression $ {a_n} = a + (n - 1)d $ and then will find the correlation between the known and the unknown terms to get the value of first term and the difference of the progression.

Complete step by step solution:
The nth term in the arithmetic progression is given by –
 $ {a_n} = a + (n - 1)d $
Where “a” is the first term and “d” is the difference between the terms.
The third term of the arithmetic expression is \[{a_3} = a + 2d\]
Similarly, sixth term of the arithmetic expression is \[{a_6} = a + 5d\]
Given that the product of the third by the sixth term of an arithmetic progression is $ 406 $
\[(a + 2d)(a + 5d) = 406\]
Expand the brackets in the above expression –
 $ {a^2} + 7ad + 10{d^2} = 406 $ …. (A)
Also, given that the division of the ninth term of the progression by the fourth term gives a quotient $ 2 $ and a remainder $ 6 $ .
 $ {a_9} = {a_4} \times 2 + 6 $
Place the standard formula-
 $ a + 8d = (a + 3d)2 + 6 $
Simplify the above expression –
 $
  a + 8d = 2a + 6d + 6 \\
  a - 2d + 6 = 0 \;
  $
 $ a = 2d - 6 $ ….. (B)
Place the above value in the equation (A)
 $ {(2d - 6)^2} + 7(2d - 6)d + 10{d^2} = 406 $
Expand the above expression –
 $ 4{d^2} - 24d + 36 + 7(2{d^2} - 6d) + 10{d^2} = 406 $
Combine the like terms and simplify the above expression –
 $ 4{d^2} - 24d + 36 + 14{d^2} - 42d + 10{d^2} = 406 $
 $
  28{d^2} - 66d + 36 = 406 \\
  14{d^2} - 33d + 18 = 203 \;
  $
Find the factors using the quadratic equation,
 $ d = \dfrac{{ - ( - 33) \pm \sqrt {{{( - 33)}^2} - 4 \times 14 \times ( - 185)} }}{{2 \times 14}} $
Simplify the above expression –
 $ d = \dfrac{{33 \pm 107}}{{28}} $
 $ d = 5,\dfrac{{ - 37}}{{28}} $
Common difference here can not be negative, so here $ d = 5 $
Place the above value in the equation (B)
 $ a = 2(5) - 6 $
 $
  a = 10 - 6 \\
  a = 4 \;
  $
Hence, first term is $ 4 $ and the common difference is $ 5 $

Note: Remember the difference between the arithmetic progression and the geometric progression and apply its concepts accordingly. In arithmetic progression, the difference between the terms remains the same whereas in the geometric progression the ratio between the terms remains the same.