AP and GP compared

The comparison between arithmetic growth and geometric growth is at the heart of many financial choices: simple interest versus compound interest.

Example — AP and GP compared

Starting salary of $1\,000$ euros a month. Two competing offers for the pay rises over $10$ years ($120$ months):

• offer A (AP): $+\,5$ euros every month.   $a_{120} = 1000 + 119\cdot 5 = 1\,595$ euros a month; total received $S_{120} = 120(1000+1595)/2 = 155\,700$ euros.
• offer B (GP): $+\,0{,}3\%$ every month ($q = 1{,}003$).   $a_{120} = 1000\cdot 1{,}003^{119} \approx 1\,427$ euros/month; total $S_{120} = 1000\cdot(1{,}003^{120}-1)/0{,}003 \approx 144\,500$ euros.

Over the $10$ years the AP pays more; but over $30$ years the situation reverses (the GP grows exponentially, the AP only linearly).

The dilemma reappears wherever simple interest (AP) and compound interest (GP) are compared.

Links

Topics: Sequences
Concepts: Arithmetic progression · Geometric progression
Methods: Sum of an arithmetic progression · Sum of a geometric progression
Skills: Model · Use formulas