All Questions ⁺⁰237697 Questions

0

32

0

+21

Algebra

Find
\sum_{k = 1}^{20} k(k^2 - 10k - 20)(k^2 + 1)

co1inqiu Mar 7, 2025

0

19

1

+810

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

crimefightingvigiI Mar 7, 2025

0

24

1

+810

Algebra

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

crimefightingvigiI Mar 7, 2025

0

15

0

+810

Algebra

Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.

crimefightingvigiI Mar 7, 2025

0

21

1

+810

Algebra

Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.

crimefightingvigiI Mar 7, 2025

0

20

0

+1285

Algebra

The closed form sum of
12 \left[ 1 \cdot 2^2 \cdot 3 + 2 \cdot 3^2 \cdot 4 + \dots + n (n + 1)^2 (n + 2) \right]
for n \ge 1 is n(n + 1)(n + 2) p(n) for some polynomial p(n). Find p(n).

AnswerscorrectIy Mar 7, 2025

0

18

0

+1285

Algebra

The sum
6 (1 \cdot 3 \cdot 5 + 2 \cdot 4 \cdot 6 + \dots + n(n + 2)(n + 4))
is equal to a polynomial f(n) for all n \ge 1.
Write f(n) as a polynomial with terms in descending order of n.

AnswerscorrectIy Mar 7, 2025

0

4

1

+1285

Algebra

A sequence a_1, a_2, a_3, \dots of positive integers has the following properties:
* The first three terms are in geometric progression.
* The second, third, and fourth terms are in arithmetic progression.
* In general, for all $i\ge1$, read more ..

AnswerscorrectIy Mar 7, 2025

Mar 6, 2025

0

27

0

+1285

Algebra

A sequence of real numbers (a_n) is defined as follows: a_0 = 1, a_2 = 2, and
a_{n + 2} = \frac{a_{n + 1}}{a_n}
for n = 0, 1, 2, \dots. Find a_0 + a_1 + a_2 + \dots + a_{100}.

AnswerscorrectIy Mar 6, 2025

0

7

1

+1285

Algebra

Let
P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81}.
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$

AnswerscorrectIy Mar 6, 2025

0

27

1

+1285

Algebra

Let
P = 2^{1/2} \cdot 4^{1/4} \cdot 8^{1/8} \cdot 16^{1/16}
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$

AnswerscorrectIy Mar 6, 2025

0

1

22

1

+190

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

0

1

14

1

+190

Algebra

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

HumanBemg Mar 6, 2025

0

14

2

+190

Algebra

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

HumanBemg Mar 6, 2025

+1

1

11

1

+190

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

Mar 5, 2025

0

1

9

1

+120

Algebra

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

coIinqiu Mar 5, 2025

0

19

1

+120

Algebra

Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.

coIinqiu Mar 5, 2025

0

1

13

1

+120

Algebra

Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.

coIinqiu Mar 5, 2025

Mar 4, 2025

0

30

1

+120

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

coIinqiu Mar 4, 2025

0

14

1

+120

Algebra

Simplify (x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1).

coIinqiu Mar 4, 2025

0

29

0

+810

Algebra

Find all real numbers a such that the roots of the polynomial
x^3 - 3x^2 + 17x + a
form an arithmetic progression and are not all real.

crimefightingvigiI Mar 4, 2025

0

28

0

+810

Algebra

Find the sum of the series
1 + \frac{1}{2} + \frac{1}{10} + \frac{1}{30} + \frac{1}{60} + ,..
where we alternately multiply by $\frac 12$ and $\frac 15$ and $\frac{1}{3}$ to get successive terms.

crimefightingvigiI Mar 4, 2025

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