Question

If \[{a_1} + {a_2} + {a_3} + {a_4} + {a_5} = 10\] and \[{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 = 50\] , then the sum of products of these numbers taken two at a time is-
A. 50
B. 25
C. 75
D. 100

Answer 1

Hint: This is based on the formula of squares of two or more terms.. First make an expression of the sum of linear terms. Then expression for sum of square terms. Further, make a relationship in these two expressions, with the help of the above said formula. Solving this relationship, we get the result as needed.

Complete step by step solution:
We have from the question,
\[{a_1} + {a_2} + {a_3} + {a_4} + {a_5} = 10\] …(1)
And
\[{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 = 50\] …(2)
Where \[{a_1},{a_2},{a_3},{a_4},{a_5}\] are some five numbers.
Since we know that for two terms ${a_1},{a_2}$ ,
$
  {({a_1} + {a_2})^2} = {a_1}^2 + {a_2}^2 + 2{a_1}{a_2} \\
   \Rightarrow {({a_1} + {a_2})^2} = {a_1}^2 + {a_2}^2 + 2({a_1}{a_2}) \\
$ …(3)
Similarly for we know that for three terms ${a_1},{a_2},{a_3}$ ,
${({a_1} + {a_2} + {a_3})^2} = {a_1}^2 + {a_2}^2 + {a_3}^2 + 2({a_1}{a_2} + {a_2}{a_3} + {a_1}{a_3})$ ..(4)
Now, above two equations (3) and (4) showing that terms written in braces on RHS is the “sum of products of these numbers taken two at a time” .
Therefore in general for 5 terms \[{a_1},{a_2},{a_3},{a_4},{a_5}\] , we will get,
${({a_1} + {a_2} + {a_3} + {a_4} + {a_5})^2} = {a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 + 2(S)$ …(6)

Where , S is “sum of products of these numbers taken two at a time” .
From the given data in the problem, we have to compute the value of S.
So, S = ?
Now, we substitute the known values from the equation (1) and (2) in equation (6) , we get
${({a_1} + {a_2} + {a_3} + {a_4} + {a_5})^2} = {a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 + 2(S)$
$ \Rightarrow {(10)^2} = 50 + 2 \times (S)$
Simplify the above expression we get
$
   \Rightarrow 100 - 50 = 2 \times (S) \\
   \Rightarrow S = \dfrac{{50}}{2} \\
   \Rightarrow S = 25 \\
$
Thus we got the value of “sum of products of these numbers taken two at a time” as 25.

So, the correct answer is “Option B”.

Note: Here, formula for algebraic expansion has been used. Such expressions are common in polynomials. Many similar kinds of results can be obtained by using many such expressions and their expansion. Actually linear algebra defines binomial expansion of the sum of two or more terms.