Problem 15 - Basic Engineering Mathematics

Question

Three numbers are in arithmetic progression. Their sum is 15 and their product is 80 . Determine the three numbers.

Step-by-Step Solution

Answer

The numbers are 2, 5, and 8.

Step 1: Understanding Arithmetic Progression

The three numbers in an arithmetic progression can be expressed as \( a-d \), \( a \), and \( a+d \), where \( a \) is the middle number and \( d \) is the common difference.

Step 2: Sum of Numbers in Arithmetic Progression

The sum of these numbers is given as 15: \[(a-d) + a + (a+d) = 3a = 15\]Solving for \( a \), we find \( a = \frac{15}{3} = 5 \).

Step 3: Product of Numbers

The product of these numbers is given as 80:\[(a-d) \, a \, (a+d) = 80\]Substitute \( a = 5 \) into the equation:\[(5-d) \times 5 \times (5+d) = 80\]

Step 4: Simplifying the Product Equation

Simplify the product equation:\[(5-d)(5+d) = 80/5 = 16\]This resembles the difference of squares:\[25 - d^2 = 16\]

Step 5: Solving for 'd'

Rearrange the equation to solve for \( d^2 \):\[d^2 = 25 - 16 = 9\]Take the square root of both sides to find \( d \):\[d = \pm 3\]

Step 6: Determining the Numbers

With \( a = 5 \) and \( d = 3 \), the numbers are \( 5-3 = 2 \), \( 5 \), and \( 5+3 = 8 \).With \( d = -3 \), the numbers remain the same: \( 2 \), \( 5 \), and \( 8 \) because the order of numbers in arithmetic progression is not affected by the sign of \( d \).

Key Concepts

Sum of Arithmetic Sequence

In arithmetic progression, the numbers are arranged in a sequence where the difference between consecutive terms is constant. For the problem at hand, we have three numbers represented as \(a-d\), \(a\), and \(a+d\). The formula for the sum of these terms is straightforward: add them together. Here, it's essential to understand how we derive the sum formula for any arithmetic sequence, which is given by \(S_n = \frac{n}{2} (2a + (n-1)d)\), where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.

However, because we are dealing with only three numbers, the sum simplifies to be directly calculated as:\[ (a-d) + a + (a+d) = 3a \]
This simplification arises because the negative and the positive \(d\) terms cancel each other out.

In our example, the total sum is 15, so when expressed as \(3a = 15\), solving for \(a\) leads to \(a = 5\). This approach is a simple and effective way to understand how arithmetic sequences can be analyzed when working with small fixed-term sequences.

Product of Numbers

The product of numbers in an arithmetic sequence often requires multiplying the terms of the sequence. For the three numbers \(a-d\), \(a\), and \(a+d\), the product formula can be understood using properties of roots and factors. That is, for a product given as \( (a-d)a(a+d) \), it can be rewritten by recognizing that it forms a classic polynomial structure. Here we substitute \(a=5\) into the product equation to reflect:

• \( (5-d) \times 5 \times (5+d) = 80 \)

This demonstrates how plugging in known values can simplify the problem's complexity. We address the product condition by simplifying further into a calculable form, deriving the equation in terms of a common base factor, such as resolving \(b\) in basic polynomial identities.

Difference of Squares

A keen insight into the problem comes from the difference of squares identity, recognized because the terms \((a-d)(a+d)\) directly relate to \(a^2 - d^2\). This mathematical identity simplifies expressions where two terms differ by a square. When applied here, we transform the multiplication:

• \( (5-d)(5+d) = 80/5 = 16 \)

After this simplification, recognizing that the structure fits the difference of squares formula \( a^2 - b^2 \), we can express and solve simply as \( 25 - d^2 = 16 \).

Solving for \(d^2\) gives us \(d^2 = 9\), leading to \(d = \pm 3\). Thus, understanding the difference of squares turns a seemingly complex multiplication into an arithmetic problem that lends itself well to simple equation balancing.