# 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

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A) G-L = 4

B) b-a = 2

C) G-L = 7

D) a-b = 2

A) G-L = 4

B) b-a = 2

C) G-L = 7

D) a-b = 2

##### 1 Answer

The correct choices are A and D.

#### Explanation:

The sum of the first

#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

For example, 11 is the

#(11+1)/2 = 6# th 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 *odd*, *even*.

We also know 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

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.