Math core problems

Problem 1 – Triadic sequence scaling#

A sequence is defined by

$$ a_n = D_3^n τ_r. $$

If $$τ_r$$ triples, how does $$a_n$$ change?

Problem 2 – Resonant integral#

Evaluate the integral

$$ I = \int_0^{τ_r} T_f D_6 , dt. $$

Express your answer in terms of $$τ_r$$.

Problem 3 – Exponential resonance equation#

Solve the equation

$$ X e^{ΛΘ t} = D_9 $$

for $$t$$.