If (1+3+5+...... +a ) + (1+3+5+ .......... +b) = (1+3+5......... +c), where each set of parentheses contains the sum of consecutive odd integers as shown such that a+b+c = 21, a>6. If G = Max{a,b,c} and L = Min{a,b,c}, then? The question has multiple ans

A) G-L = 4
B) b-a = 2
C) G-L = 7
D) a-b = 2

1 Answer
Feb 4, 2018

a = 7" "" "b=5" "" "c=9

The correct choices are A and D.

Explanation:

The sum of the first n odd numbers is n^2.

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

...and so on.

A number's position within the set of odd numbers is (n+1)/2

For example, 11 is the (11+1)/2 = 6th odd number

Using these two facts, we can rewrite our expression as:

(1+3+5+cdots+a) + (1+3+5+cdots+b) = (1+3+5+cdots+c)

"sum of first "(a+1)/2" odd numbers" + "sum of first "(b+1)/2" odd numbers" = "sum of first "(c+1)/2" odd numbers"

((a+1)/2)^2+((b+1)/2)^2 = ((c+1)/2)^2

Doing a little bit of algebra to this, we can simplify the expression:

(a+1)^2/4+(b+1)^2/4 = (c+1)^2/4

(a+1)^2+(b+1)^2 = (c+1)^2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Since a, b, and c are all odd, (a+1), (b+1), and (c+1) must all be even.

We also know that a+b+c = 21, which means that:

(a+1)+(b+1)+(c+1)
= a+b+c+3
= 21+3
= 24

There's only one Pythagorean triple that adds up to 24:

6^2+8^2 = 10^2

6 + 8 + 10 = 24

Since we know that a>6, we also know that a+1>7.

Therefore, we know that a+1 must be 8 since it can't be 6.

This means that b+1 is 6 and c+1 is 10.

Therefore, our three numbers are:

a = 7 " "" "b=5 " "" "c=9

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using these numbers, we can figure out which of the 4 statements are true.

G = max{a,b,c} = max{7,5,9} = 9

L = min{a,b,c} = min{7,5,9} = 5

Statement A

G - L = 9 - 5 = 4" "" "" "" " So, statement A is true.

Statement B

b - a = 5 - 7 = -2" "" "" " So, statement B is false.

Statement C

G - L = 9 - 5 = 4" "" "" "" "So, statement C is false

Statement D

a - b = 7 - 5 = 2" "" "" "" " So, statement D is true.