The sum of the first term of an AP is 15 and their product is 80 . Find the AP
ScanSolve AI Answer
Step-by-step solution
7 stepsLet's break this down. The phrasing "The sum of the first term of an AP is 15 and their product is 80" is a bit unusual. Given "their product," it's most likely referring to the first two terms of the Arithmetic Progression (AP). We will proceed with this interpretation.
Define the terms of the AP.
Let the first term of the AP be a₁.
Let the second term of the AP be a₂.
In an Arithmetic Progression, the second term can be expressed as a₂ = a₁ + d, where d is the common difference.Set up equations based on the given information.
From the problem, we have two conditions:
- The sum of the first two terms is 15: a₁ + a₂ = 15
- The product of the first two terms is 80: a₁ × a₂ = 80
Substitute a₂ = a₁ + d into the sum equation.
a₁ + (a₁ + d) = 15
2a₁ + d = 15
From this, we can express the common difference d in terms of a₁:
d = 15 - 2a₁Substitute a₂ = a₁ + d into the product equation.
a₁ × (a₁ + d) = 80
Now, substitute the expression for d (from Step 3) into this equation:
a₁ × (a₁ + (15 - 2a₁)) = 80
a₁ × (15 - a₁) = 80
15a₁ - a₁² = 80Rearrange the equation into a standard quadratic form (Ax² + Bx + C = 0).
a₁² - 15a₁ + 80 = 0
Solve the quadratic equation for a₁ using the quadratic formula.
The quadratic formula is x = (-B ± √(B² - 4AC)) / (2A).
In our equation, A = 1, B = -15, and C = 80.
First, calculate the discriminant (Δ = B² - 4AC):
Δ = (-15)² - 4(1)(80)
Δ = 225 - 320
Δ = -95Analyze the result.
Since the discriminant (Δ) is -95, which is a negative number, there are no real solutions for a₁. This means that there are no real numbers for the first term (and consequently, no real common difference) that satisfy the given conditions.
Therefore, based on the most plausible interpretation of the problem statement, there is no Arithmetic Progression with real terms that satisfies the given conditions.
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