Algebra

If $P(n)$ is $2n + 1 < 2^n$, what is the smallest natural number $n$ for which the base case $P(n)$ is true?

If $P(n)$ is $1 + \frac{1}{4} + \frac{1}{9} + \dots + \frac{1}{n^2} < 2 - \frac{1}{n}$, then $P(2)$ is:

The statement $P(k) \implies P(k+1)$ is called the:

If $P(n)$ is $3^{2n+2} - 8n - 9$ is divisible by 64, what is the value of $P(1)$?

In the Principle of Mathematical Induction, the assumption that the statement $P(n)$ is true for $n=k$ is known as the:

The expression $x^n + y^n$ is divisible by $x+y$ if and only if:

The expression $a^n - b^n$ is divisible by $a+b$ if and only if:

In the induction step for the statement $P(n) = n^3 + 2n$ is divisible by 3, what is the expansion of $(k+1)^3 + 2(k+1)$?

The sum $\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + ... + \frac{1}{(3n-2)(3n+1)}$ is equal to:

If $P(n)$ is the statement $n! > 2^n$, and we assume $P(k)$ is true for $k \ge 4$, how is $P(k+1)$ proven?

If $x + iy = i$, find the value of the real part $x$.

For the general quadratic equation $ax^2 + bx + c = 0$, what is the formula for the sum of the roots?

If one root of the quadratic equation $x^2 + 9 = 0$ is $3i$, what is the other root?

What is the value of $i^3$?

Find the value of $k$ if the difference between the roots of the equation $x^2 - kx + 12 = 0$ is 1.

If $x = a+b, y = a\omega + b\omega^2, z = a\omega^2 + b\omega$, then find the value of $x^3 + y^3 + z^3$.

Find the roots of the quadratic equation $x^2 + \sqrt{2}x + 1 = 0$.

If one root of the quadratic equation $x^2 - 4x + k = 0$ is $2 + i$, find the value of the constant $k$.

If $\tan A$ and $\tan B$ are the roots of $x^2 + px + q = 0$, find the value of $\sin^2(A + B)$ in terms of $p$ and $q$.

If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find the value of $\alpha^{30} + \beta^{30}$.

If $x \in \{0, 1, 2, 3, 4, 5\}$, what is the solution for $2x - 3 > 4$?

What is the solution of $\frac{x}{3} + 4 > 4$ for $x \in R$?

Solve $1 - x < 2$.

Solve $5x + 2 \le 17$ for $x \in N$.

Find the solution set for $x$: $-5 \le \frac{2 - 3x}{4} \le 9$.

Find the range of $x$: $\frac{5x - 2}{3} - \frac{7x - 3}{5} > \frac{x}{4}$.

Solve the inequality: $4x + 3 < 6x + 7$.

How many positive integers $n$ satisfy $\frac{n}{20} < \frac{4}{n}$?

Find the range of $x$ for which $||x|-1| < 1$.

Solve the inequality: $\frac{x^2 - x - 6}{x-1} \ge 0$.

Solve for $n$ if $\frac{n!}{(n-2)!} = 6$.

How many 5-digit numbers can be formed using the digits $\{1, 2, 3, 4, 5\}$ such that each digit is used exactly once?

Calculate the value of $^5C_2$.

How many ways can the letters of the word 'SUN' be rearranged?

How many arrangements of the letters of the word 'PRIME' begin with a vowel?

A basket contains 3 identical apples, 4 identical oranges, and 2 identical mangoes. In how many ways can at least one fruit be selected?

If $^nC_2 = 45$, then find the value of $n$.

What is the exponent of 3 in the prime factorization of $100!$?

In how many ways can 7 people be seated around a circular table such that two particular people are separated by at least 2 other people?

How many 3-digit numbers have the property that the sum of their digits is an even number?

What is the second term in the expansion of $(x + 2)^4$?

The coefficient of $x^2$ in the expansion of $(1 - x)^3$ is:

The value of $^nC_r / ^nC_{n-r}$ is:

Which term in the expansion of $(1 + x)^n$ has the coefficient $^nC_r$?

The sum of the last $n$ coefficients in $(1+x)^{2n-1}$ is:

Find the coefficient of $x^7$ in $(a - bx)^{10}$.

In the expansion of $(1+x)^n$, the ratio of the coefficients of $x^r$ and $x^{r-1}$ is:

If $n$ is an even natural number, the value of the sum $\sum_{r=0}^{n} \frac{(-1)^r}{\binom{n}{r}}$ is equal to:

The total number of terms in the expansion of $(x+y+z)^{10}$ is:

The coefficient of $x^5$ in the expansion of $(1+x+x^2+x^3)^{10}$ is:

What is the common ratio of the GP sequence: $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \dots$?

Find the $7^{th}$ term of the AP: $7, 14, 21, \dots$.

Find the $4^{th}$ term of a GP where $a = 1$ and $r = 10$.

Calculate the sum of the first $5$ odd natural numbers ($1, 3, 5, 7, 9$).

Find the sum of $n$ terms of the sequence $0.7, 0.77, 0.777, \dots$

If the sum of the first $n$ terms of an AP is $S_n = n^2 + 2n$, find the common difference.

Find the sum of all integers from 1 to 100 that are divisible by either 2 or 5.

What is the sum of $n$ terms of the series $1 + (1+2) + (1+2+3) + \dots$?

If $a, b, c$ are in AP and $a, b, d$ are in GP, then $a, a-b, d-c$ are in... (Wait, this was excluded. Replacement: If $a, b, c$ are in GP, then $\log_a x, \log_b x, \log_c x$ are in:)

The sum of the first 24 terms of the AP $a_1, a_2, a_3, \dots$ if it is known that $a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225$ is: