If $S_{n} = {(4)}^{n} + 3$ , what is the units digit of $S_{100}$ ?

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Solution

Begin by listing the first few terms of the sequence in order to find the pattern:
$S_{1} = 4^{1} + 3 = 4 + 3 = 7$
$S_{2} = 4^{2} + 3 = 16 + 3 = 19$
$S_{3} = 4^{3} + 3 = 64 + 3 = 67$
$S_{4} = 4^{4} + 3 = 256 + 3 = 259$
The units digit of all odd-numbered terms is $7$ .
The units digit of all even-numbered terms is $9$ .
Because $S_{100}$ is an even-numbered term, its units digit will be $9$ .

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