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Algebra 2 9-12 / Week 06 / Thursday
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Week 06 · Exponential Functions and Geometric Series

Thursday

The sum of the series
// Duckweed doubling across the trough
⏱ about 20 min

Thursday: The Sum of the Series

"Yesterday the totals were 3, 9, 21, 45, 93, 189," Wren says, writing them on the chalkboard wall.

"Each one is three less than the next pile," Comet says. "45 is 48 minus 3. 93 is 96 minus 3."

"Then there is a formula," Wren says. "Call the sum S. Write out the terms. What do you notice if I double S?"

Comet writes 2S under S. "Every term shifts one place right. Six, twelve, twenty-four, up to 192."

Nova dims the matching terms. "Would you like a hint? Subtract S from 2S."

"Almost everything cancels," Comet says. "2S minus S is 192 minus 3. So S is 189."

"And with any ratio r, the same trick gives S = a(rⁿ - 1)/(r - 1)," Wren says. "Today we prove it."

The theorem

A geometric series adds the first n terms of a geometric sequence: S = a + ar + ar² + ... + arⁿ⁻¹.

When the ratio r is not 1, the sum is S = a(rⁿ - 1)/(r - 1). When r = 1, every term is a and the sum is just n × a.

For the beans: a = 3, r = 2, n = 6. S = 3(2⁶ - 1)/(2 - 1) = 3 × 63 = 189. It matches the lab.

The derivation, read first

  1. Write the sum: S = a + ar + ar² + ... + arⁿ⁻¹.
  2. Multiply every term by r: rS = ar + ar² + ... + arⁿ⁻¹ + arⁿ. Each term moves one place right.
  3. Subtract the first line from the second. Every middle term cancels: rS - S = arⁿ - a.
  4. Factor both sides: S(r - 1) = a(rⁿ - 1).
  5. Divide by r - 1, which is allowed because r is not 1: S = a(rⁿ - 1)/(r - 1).

Every step is a move you already own: multiply both sides, subtract, factor, divide. The only care is r ≠ 1, so the division is legal.

Yesterday's pattern falls out of step 3 with r = 2. There 2S - S = S, so S = 3 × 2ⁿ - 3. Each total is 3 less than the next pile.

Using the formula

SeriesarnSum
3 + 6 + 12 + 24 + 48 + 96326189
2 + 6 + 18 + 54 + 162235242
16 + 8 + 4 + 2 + 1161/2531
SUM THE SERIES
  • Read the question.
  • Tap your answer.
A geometric series starts at 3 with ratio 2. What is the sum of the first 6 terms?
A geometric series starts at 2 with ratio 3. What is the sum of the first 5 terms?
A geometric series starts at 4 with ratio 3. What is the sum of the first 4 terms?
A geometric series starts at 1 with ratio 2. What is the sum of the first 10 terms?
THE DERIVATION, IN ORDER
  • ?Divide by r - 1, with r not 1: S = a(rⁿ - 1)/(r - 1)
  • ?S = a + ar + ar² + ... + arⁿ⁻¹
  • ?Subtract: rS - S = arⁿ - a, because the middle terms cancel
  • ?Factor: S(r - 1) = a(rⁿ - 1)
  • ?rS = ar + ar² + ... + arⁿ⁻¹ + arⁿ
WHY THIS EXERCISEA derivation is a chain of legal moves. Once you can rebuild it, you never need to memorize the formula.
In the derivation, what do you multiply S by to shift every term one place? Type the letter.
The formula S = a(rⁿ - 1)/(r - 1) is not allowed for one value of r. Type that number.
What is 3 + 6 + 12 + 24 + 48 + 96? Type the number.
StatementTrue or false?
Multiplying S by r shifts every term of the series one place.?
3 × (64 - 1) ÷ (2 - 1) = 189?
The formula a(rⁿ - 1)/(r - 1) works when r = 1.?
2 × (243 - 1) ÷ (3 - 1) = 242?
When r = 1, the sum of n terms is n × a.?
WHY THIS EXERCISEKnowing when a formula is allowed is part of knowing the formula.
Try it
Write 1 + 2 + 4 + 8 + 16 on a card. Add it by hand, then use the formula with a = 1, r = 2, n = 5.
Do the two answers agree? Write one sentence about which way was quicker.

Sharp proving. Tomorrow the formula plans a watering schedule and the week comes together.

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