Question

Sum of of the squares of 3 consequitive natural number is 677. Determine the largest number of the three

Answer 1

Answer=16

square of 14=196

square of 15=225

square of 16=256

196+225+256=677

Hence the largest number of the three numbers is 16.

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Answer 2

Answer:

Required largest number is 16.

Step-by-step explanation:

Given,sum of of the squares of 3 consequitive natural number is 677

But now question is what is natural number ?

all the positive integers from 1 till infinity and are also used for counting purpose are called natural number.

For example, 2,3,4,5 e.t.c all are natural numbers .

Let three consecutive numbers be x,(x+1),(x+2).

Now square of x $= {x}^{2}$

and square of (x+1) $= {(x + 1)}^{2}$

and square of (x+2) $= {(x + 2)}^{2}$

According to questions,

${x}^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} = 677$

${x}^{2} + {x}^{2} + 2x + {1}^{2} + {x}^{2} + 2 \times x \times 2 + {2}^{2} = 677$

${x}^{2} + {x}^{2} + 2x + 1 + {x}^{2} + 4x + 4 = 677$

$3 {x}^{2} + 6x + 5 = 677 \\ 3 {x}^{2} + 6x + 5 - 677 = 0 \\ 3 {x}^{2} + 6x - 672 = 0 \\ 3 {x}^{2} + (48 - 42)x - 672 = 0 \\ 3 {x}^{2} + 48x - 42x - 672 = 0 \\ 3x(x + 16) - 42(x + 16) = 0 \\ (x + 16)(3x - 42) = 0$

If product of two terms is zero then they are separately zero.

So,

$x + 16 = 0 \\ x = - 16$

and

$3x - 42= 0 \\ 3x = 42\\ x = \frac{42}{3} \\ x = 14$

$x = - 16$

is not possible (because x is natural number and natural number is always positive)

So, value of x is 14.

Therefore, three consecutive numbers are 14,15,16.

Hence, required largest number is 16.

Natural number related two more questions:

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