# Question #fcc37

Dec 7, 2017

$53$

#### Explanation:

let$\text{ "n" }$be the middle number

then we have
1st number$= n - 2$

2nd number$= n - 1$

4th number $n + 1$

5th number$= n + 2$

$\left(n - 2\right) + \left(n - 1\right) + n + \left(n + 1\right) + \left(n + 2\right) = 270$

$n - \cancel{2} + n - \cancel{1} + n + n + \cancel{1} + n + \cancel{2} = 270$

$5 n = 270$

$n = \frac{270}{5} = 54$

$2 n d \text{ number } = 54 - 1 = 53$

Dec 7, 2017

Entire sequence: $52 , 53 , 54 , 55 , 56$
Second number in the sequence: $53$

#### Explanation:

If the sum of consecutive integers is equal to $270$, we can write the first integer as $x$, the 2nd as $x + 1$ (because they are consecutive), etc.

• First integer: $x$
• Second integer: $x + 1$
• Third integer: $x + 2$
• Fourth integer: $x + 3$
• Fifth integer: $x + 4$

Notice with each integer, you are increasing the number by $1$. If that seems confusing, think about it this way:

If the first integer was $x$, the second will be $x + 1$. With the third integer, we still increase by one from the previous integer:

$\left(x + 1\right) \underline{+ 1} = x + 2$

Let's set up our equation:

$x + \left(x + 1\right) + \left(x + 2\right) + \left(x + 3\right) + \left(x + 4\right) = 270$

We have a good deal of terms to combine, so let's do that.

$x + x + x + x + x = 5 x$

$1 + 2 + 3 + 4 = 10$

From this, we can make our new equation

$5 x + 10 = 270$

Subtract $10$ from both sides to get

$5 x = 260$

Divide both sides by $5$ to get

$x = 52$

This isn't our answer, however. We found what $x$ is (the first integer), but we want to know the second number in this sequence is, and to do that, just plug $52$ into our expression for the second integer, $\left(x + 1\right)$

$52 + 1 = 53$

The second integer is $53$.