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Home » You Won’t Believe What Makes Ponchatoula Louisiana the Best Keepsake Town! - Sterling Industries

You Won’t Believe What Makes Ponchatoula Louisiana the Best Keepsake Town! - Sterling Industries

📅 April 19, 2026 👤 scraface

Apr 19, 2026

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The series \(\sum_{n=1}^{50} \frac{1}{n(n+1)}\) is a telescoping serie...
Partial fraction decomposition:
\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}
Write the series using partial fractions:
\sum_{n=1}^{50} \left( \frac{1}{n} - \frac{1}{n+1} \right)
Observe the telescoping nature:
The series expands as:
Simplify the result:
A science policy analyst is reviewing a proposal for a rectangular sol...
To maximize the area of a rectangle with a fixed perimeter, the rectan...

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a_7 = a_6 + a_5 = 21 + 13 = 34 \\
a_8 = a_7 + a_6 = 34 + 21 = 55
So, number of 8-digit numbers **without** consecutive 3s is 55.
Thus, the number **with** at least one pair of consecutive 3s is:
- 55 = \boxed{201}
\frac{a + 2b}{a - 2b} + \frac{a - 2b}{a + 2b} = 2.
Find the value of $ \frac{a^2 + 4b^2}{a^2 - 4b^2} $.
Let’s simplify the left-hand side. Let $ x = \frac{a + 2b}{a - 2b} $....
x + \frac{1}{x} = 2 \Rightarrow x^2 - 2x + 1 = 0 \Rightarrow (x - 1)^2...
\frac{a + 2b}{a - 2b} = 1.

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