HumanBemg

18 Questions
2 Answers

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+75

Algebra

Find x if \log_2 x^2 + \log_{1/2} x + 3 \log_4 x = 5.

HumanBemg Mar 8, 2025

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Algebra

Simplify 25^{\frac{1}{2} - \log 5 + \sqrt{3}}.

HumanBemg Mar 8, 2025

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+75

Algebra

Compute
\log_4 5 + \log_5 6 + \log_6 7+ \log_{2047} 2048 + \log_{2048} 2049

HumanBemg Mar 8, 2025

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+75

Algebra

Fill in the blanks, to make a true equation.
3/(3^2 - 1) + 3^2/(3^4 - 1) + 3^3/(3^6 - 1) + 3^4/(3^8 - 1) + ... + 3^(2(n - 1))/(3^(2n) - 1) = ___/___

HumanBemg Mar 8, 2025

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Counting

Let S be the set {1, 2, 3, \dots, 10, 11, 12}. How many subsets of the set S have no two consecutive primes as members?

HumanBemg Mar 7, 2025

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Algebra

Let
A_0 = 0
A_1 = 1
A_n = A_{n - 1} + A_{n - 2} for n ge 2
There is a unique ordered pair (c,d) such that c \alpha^n + d \beta^n is the closed form for sequence A_n. Find the ordered pair (c,d).

HumanBemg Mar 7, 2025

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Algebra

Find the ordered pair (p,q) such that
F_n = p \alpha^n + q \beta^n.

HumanBemg Mar 7, 2025

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Algebra

The Fibonacci sequence, is defined by F_0 = 0, F_1 = 1, and F_n = F_{n - 2} + F_{n - 1}. It turns out that
F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}},
where \alpha = \frac{1 + \sqrt{5}}{2} and \beta = \frac{1 - \sqrt{5}}{2}.
The read more ..

HumanBemg Mar 7, 2025

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Algebra

Find a closed form for
S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]
for any integer n \ge 1. Your response should have a factorial.

HumanBemg Mar 7, 2025

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Algebra

For a positive integer k, let
S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).
Find a closed form for S_k.

HumanBemg Mar 7, 2025

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+75

Algebra

Let r be a real number such that |r| < 1. Express
\sum_{n = 0}^{\infty} n*r^n*(n + 1)*(n + 2)
in terms of r.

HumanBemg Mar 6, 2025

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Algebra

Find
\sum_{k = 0}^{10} (k + 3) \cdot 2^k \cdot (k - 3)

HumanBemg Mar 6, 2025

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2

+75

Algebra

Simplify \frac{1 + 3 + 5 + ... + 1999 + 2001 + 2003}{2 + 4 + 6 + ... + 2000 + 2002 + 2004 + 2006 + 2008 + 2010}.

HumanBemg Mar 6, 2025

+1

1

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+75

Algebra

Find the sum
\frac{1}{7} + \frac{2}{7^2} + \frac{3}{7^3} + \frac{1}{7^4} + \frac{2}{7^5} + \frac{3}{7^6}

HumanBemg Mar 6, 2025

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+75

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 7n - 3mn + 15m + 8n + ___

HumanBemg Mar 3, 2025

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+75

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

HumanBemg Mar 3, 2025

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+75

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

HumanBemg Mar 3, 2025

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+75

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

HumanBemg Mar 3, 2025

+75

0

(A) k > 0 and a > 1

(B) k < 0 and a < 1

(C) k < 0 and a > 1

(D) k > 0 and a < 1

HumanBemgMar 8, 2025