Math Solutions

Find the 8th term of 2, 6, 18, 54, ... where r = 3.

Find the sum of the first 100 natural numbers: S = 100(101)/2 = 5050.

Find the 50th term of 3, 7, 11, 15, ... with common difference d = 4.

Show that lim(n→∞) (1+1/n)^n = e ≈ 2.718. Animate the value converging.

Evaluate log₅(17) using change of base: ln(17)/ln(5).

A bacteria colony starts at 500 and doubles every 3 hours. How many after 12 hours?

Convert 5³=125 to log form and log₄(64)=3 to exponential form.

Expand log₃(x²y/z³) using product, quotient, and power rules.

Solve log₂(x) + log₂(x−2) = 3. Combine logs, convert to exponential, solve, check domain.

Solve 3^(2x−1) = 81. Rewrite 81 as 3⁴ and equate exponents: 2x−1 = 4, x = 5/2.

Compute 3√8 + 2√18 − √50. Simplify each radical, then combine like terms.

Simplify 5/(√3−1) by multiplying by the conjugate (√3+1)/(√3+1).

Solve √(2x+3) = x−1. Square both sides and check for extraneous roots.

Simplify √(72x⁵y³). Factor under the radical and extract perfect squares.

Graph f(x) = (2x+1)/(x−3). Show vertical asymptote at x=3 and horizontal asymptote at y=2.

Solve 1/x + 1/(x+2) = 5/8. Clear denominators and check for extraneous solutions.

Multiply (x²−4)/(x+3) · (x+3)/(x−2). Show factor cancellation.

Add 2/(x+1) + 3/(x−2). Find the common denominator and combine.

Simplify (x²−9)/(x²+5x+6). Factor and cancel common factors.

Find all zeros of f(x) = x⁴−5x²+4. Show the graph crossing the x-axis at each zero.