Series

The challenge at the boards was blunt: add every whole number from 1 to 100, no calculator, five minutes. The groups that finished early all rediscovered the same trick — pair the ends.¹ A series is what you get when you add the terms of a sequence, and both families from Sequences and Their Rules have a shortcut that sidesteps the adding entirely.

Adding without adding

Pair 1 with 100, 2 with 99, 3 with 98: fifty pairs, each summing to 101, total $50\times 101=5050$. The pairing works for any arithmetic series because the gaps are even — first-plus-last equals second-plus-second-last, all the way in. In general, $n$ terms make $\frac{n}{2}$ pairs:

$$S_n=\frac{n(t_1+t_n)}{2}\text{or, expanding }{t}_n,S_n=\frac{n}{2}\left[2a+(n-1)d\right]$$

So $3+7+11+\cdots$ taken to 40 terms is $S_{40}=\frac{40}{2}\left[2(3)+39(4)\right]=20(162)=3240$ — four numbers multiplied, not forty added.

Multiply, shift, subtract

Geometric series need a different trick, and it is a beautiful one. Write the sum, multiply the whole line by $r$, and subtract: every term but two cancels, leaving

$$S_n=\frac{a\left(r^n-1\right)}{r-1}$$

For $2+6+18+\cdots$ to 10 terms: $S_{10}=\frac{2\left(3^{10}-1\right)}{3-1}=3^{10}-1=59,048$. Reproduce the cancellation yourself once on paper — write $S$, write $3S$ beneath it, subtract — rather than accepting the formula on faith; the derivation is the understanding, and it is worth a page in your Math Journal.

Where this earns its living: a savings plan with a regular deposit is a geometric series in disguise — every deposit grows for a different length of time, and the total is exactly this sum. That story is told properly in Money Over Time, and Sequences, Series, and Interest Practice runs the full route from formula to future value.

Curriculum connection

C2.3

determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate the sum of a given number of consecutive terms Sample problem: Given the following array built with grey and white connecting cubes, investigate how different ways of determining the total number of grey cubes can be used to evaluate the sum of the arithmetic series $1+2+3+4+5$. Extend the series, use patterning to make generalizations for finding the sum, and test the generalizations for other arithmetic series. [The document prints a diagram here: a 5-row staircase-pattern array of grey and white connecting cubes.]

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C2.4

solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

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Footnotes

1. The trick is usually credited to ten-year-old Carl Friedrich Gauss, whose teacher assigned the sum to keep the class busy. The story has surely been polished by two centuries of retelling — but the mathematics is genuine, and rediscovering it at a whiteboard needs no permission from history. ↩