Consecutive Multiples of 7 Explained

The problem asks us to identify the smallest number from a set of three consecutive multiples of 7, given that their sum equals 840. We will use algebraic methods to solve this.

Algebraic Setup for Multiples of 7

To represent three consecutive multiples of 7 algebraically:

• Let the smallest multiple be represented as $7n$. Here, $n$ stands for an integer.
• The next consecutive multiple of 7 would be $7n + 7$.
• The third consecutive multiple of 7 would be $7n + 14$.

Equation for Sum of Multiples

According to the question, the sum of these three consecutive multiples is 840. We can express this as:

$$7n + (7n + 7) + (7n + 14) = 840$$

Solving for 'n' in the Multiple Equation

Let's solve the equation step-by-step:

1. Combine all terms containing $n$: $7n + 7n + 7n = 21n$.
2. Combine the constant terms: $7 + 14 = 21$.
3. The equation simplifies to: $21n + 21 = 840$.
4. To isolate the term with $n$, subtract 21 from both sides: $21n = 840 - 21$.
5. This results in: $21n = 819$.
6. Now, divide both sides by 21 to find the value of $n$: $n = \frac{819}{21}$.
7. Performing the division gives us: $n = 39$.

Finding the Smallest Multiple of 7

We initially represented the smallest multiple as $7n$. Substitute the value of $n=39$ that we found:

Smallest multiple = $7 \times 39$

Calculating this product:

Smallest multiple = 273

Verifying the Consecutive Multiples Sum

To confirm our result, let's determine the other two multiples and check if their sum is 840:

• The smallest multiple is 273.
• The second multiple is $273 + 7 = 280$.
• The third multiple is $280 + 7 = 287$.

Now, let's sum these three numbers: $273 + 280 + 287$.

The sum is $553 + 287 = 840$.

Since the sum matches the value given in the problem, our calculated smallest multiple is correct.

Conclusion: Smallest Multiple Found

The smallest of the three consecutive multiples of 7 that add up to 840 is 273.