Question

The sum of 3 numbers in AP is 18. If the product of the first and third number is 5 times the common difference, find the numbers?

Answer 1

Hint: The given question is of arithmetic series in which we have to understand the formulae, related with the given question. Here some conditions are given in the question, which we have to write and then solve accordingly to get the solution for the question.

Formula used:
Common difference of AP can be obtained as the difference between the two consecutive numbers.
The consecutive terms of AP can be written as if the first term is “a” and the common difference is “d” then:
\[terms = a,a + d,a + 2d,a + 3d...\]

Complete step-by-step answer:
Here the given question is of arithmetic series, in order to solve the given question here we need to write the conditions in the mathematical form first:
Let the common difference of the series be “d”, and the first term be “a”,
Now the condition given are:
\[ \Rightarrow a - d + a + a + d = 18\,\]
And
\[ \Rightarrow (a-d) \times (a + d) = 5d\]
Solving the first condition we get:
\[ \Rightarrow (a - d) + (a) + (a + d) = 18\, \\
   \Rightarrow 3a = 18 \\
   \Rightarrow a = \dfrac{{18}}{3} = 6 \]
Here we get the value of the first term after solving the first condition.
Now using the value of “a”, in the second condition we get:
\[ \Rightarrow (a - d) \times (a + d) = 5d \\
   \Rightarrow ({a^2} - {d^2}) = 5d \\
   \Rightarrow 36 - {d^2} = 5d \\
   \Rightarrow {d^2} + 5d - 36 = 0 \\
   \Rightarrow {d^2} + (9 - 4)d - 36 = 0 \\
   \Rightarrow {d^2} + 9d - 4d - 36 = 0 \]
\[ \Rightarrow d(d + 9) - 4(d + 9) = 0 \\
   \Rightarrow (d + 9)(d - 4) = 0 \\
   \Rightarrow d = - 9,4 \]
Now we obtained the value of common difference, by using the value of first term and the second condition:
For common difference to be “-9”
\[
   \Rightarrow (a - d) = 6 - ( - 9) = 15 \\
   \Rightarrow a = 6 \\
   \Rightarrow (a + d) = 6 + ( - 9) = - 3 \]
For common difference to be “4”
\[ \Rightarrow (a - d) = 6 - (4) = 2 \\
   \Rightarrow a = 6 \\
   \Rightarrow (a + d) = 6 + (4) = 10 \]

Note: In order to solve the question of arithmetic progression we need to use the direct formulae, of the progression, we know the formulae for the first term, last terms, common difference and summation of the series also, according to the given condition in the question we can use the formulae and solve the question.