# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Discrete exponential growth: proving x_n=x_0·r^n from x_(n+1)=r·x_n by induction](/en/derivations/derivation-mko-discrete-exponential-growth-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-discrete-exponential-growth)

In the model x_(n+1)=r·x_n with a fixed time step and constant factor r≥0, prove by induction that x_n=x_0·r^n for every natural number n, and explain that the recurrence and the closed form describe the same sequence; with x_0=8 and r=3/2 compute x_1 to x_4 and compare with the closed form; describe the behaviour for r>1, r=1, 0≤r<1 and x_0=0; decide that the data 8, 12, 18, 26 fit no fixed-factor model starting at x_0=8; and explain why the per-step factor 3/2 is not the continuous growth rate 1/2.

Discrete exponential growth model

derivation-mko-discrete-exponential-growth-01





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