Question

Find the three numbers in A.P whose sum is 15 and whose product is 105.

Answer 1

Hint: If three numbers are in arithmetic series then the numbers will have a common difference.
In this question since the three numbers are in Arithmetic progressive series so we will assume three numbers which have the same difference and as their sum and product are given so we will form equations and by solving those equations we will find the three numbers.

Complete step-by-step answer:
Let the three numbers which are in A.P be \[\left( {a - d} \right),a,\left( {a + d} \right)\] respectively, these numbers are in AP since their common difference is \[d\] .
Now it is given that the sum of the numbers in A.P. is 15, hence we can write this as
 \[\left( {a - d} \right) + a + \left( {a + d} \right) = 15 - - (i)\]
Therefore by solving this equation we get
 \[
  \left( {a - d} \right) + a + \left( {a + d} \right) = 15 \\
   \Rightarrow a + a + a = 15 \\
   \Rightarrow 3a = 15 \\
   \Rightarrow a = 5 \;
 \]
Hence we get \[a = 5\]
Now it is also said that the product of the A.P numbers is 105, hence we can write this as
 \[\left( {a - d} \right) \times a \times \left( {a + d} \right) = 105\]
Now this equation can be further as
 \[a \times \left( {{a^2} - {d^2}} \right) = 105\left\{ {\because \left( {{x^2} - {y^2}} \right) = \left( {x + y} \right)\left( {x - y} \right)} \right\}\]
Here since we already know \[a = 5\] , hence we can write this equation as
 \[
  5 \times \left( {{5^2} - {d^2}} \right) = 105 \\
   \Rightarrow \left( {25 - {d^2}} \right) = \dfrac{{105}}{5} \\
   \Rightarrow \left( {25 - {d^2}} \right) = 21 \;
 \]
By further solving this, we get
 \[
  \left( {25 - {d^2}} \right) = 21 \\
   \Rightarrow {d^2} = 25 - 21 \\
   \Rightarrow {d^2} = 4 \\
   \Rightarrow d = \pm 2 \;
 \]
Hence we get the value of \[d\] which is the common difference of this series \[ = \pm 2\]
Therefore we get the values \[a = 5\] and \[d = \pm 2\]
Now if the common difference in the series is \[d = - 2\] , the series will be \[7,5,3\]
Also when \[d = 2\] , the series will be \[3,5,7\]
Therefore, the three numbers which are in A.P whose sum is 15 and whose product is 105 is \[3,5,7\]
So, the correct answer is “3,5,7”.

Note: In an arithmetic series if the common difference is \[d > 0\] then the series is progressively increasing series and if the common difference is \[d < 0\] then the series is progressively decreasing. The common difference of a series determines the nature of a series and can be used to find the missing terms of the series if the next number or the previous number of missing terms is known to us.