Three consecutive positive integers a, b and c are such that b2

Question

Three consecutive positive integers a, b and c are such that b2
 = 4(a + c). Find the value of c

GPT-4o mini · Answer

Let the three consecutive positive integers be $a$ , $b$ , and $c$ . Since they are consecutive integers, we can express them as:

$a = n, b = n + 1, c = n + 2$

for some positive integer $n$ .

According to the problem, we have the equation:

$b^{2} = 4 (a + c)$

Substituting $a$ , $b$ , and $c$ in terms of $n$ :

$(n + 1)^{2} = 4 (n + (n + 2))$

This simplifies to:

$(n + 1)^{2} = 4 (2 n + 2)$

Calculating the right side:

$(n + 1)^{2} = 8 n + 8$

Expanding the left side:

$n^{2} + 2 n + 1 = 8 n + 8$

Rearranging this equation gives:

$n^{2} + 2 n + 1 - 8 n - 8 = 0$

Simplifying further:

$n^{2} - 6 n - 7 = 0$

Now, we can solve this quadratic equation using the quadratic formula:

$n = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

In this case, $a = 1$ , $b = - 6$ , and $c = - 7$ :

$n = \frac{6 \pm \sqrt{(- 6)^{2} - 4 \cdot 1 \cdot (- 7)}}{2 \cdot 1}$

Calculating the discriminant:

$\sqrt{36 + 28} = \sqrt{64} = 8$

Thus:

$n = \frac{6 \pm 8}{2}$

Calculating the two possible values for $n$ :

$n = \frac{14}{2} = 7 or n = \frac{- 2}{2} = - 1$

Since $n$ must be a positive integer, we take $n = 7$ .

Now, substituting back to find $c$ :

$c = n + 2 = 7 + 2 = 9$

Thus, the value of $c$ is:

$9$