Applications of Geometric Series

What a Repeating Decimal Really Is

🔁 A Repeating Decimal Is an Infinite Geometric Series

①0.777… is 7/10 + 7/100 + 7/1000 + …

②First term 7/10, ratio 1/10, an infinite geometric series

③Since |1/10|<1 it converges, and its sum is the fraction form

Example 1

Express the repeating decimal 0.777… as a reduced fraction.

View it as an infinite geometric series with first term 7/10 and ratio 1/10.

Apply the sum formula S = a/(1−r).

S = 7/101 - 1/10 = 7/109/10

▸ 7/9

One repeating digit gives denominator 9, two gives 99 — it falls out naturally from the geometric series.

Repeating Decimal → Fraction Formula

Converting a repeating decimal

0.aaa… = a/9 , 0.ababab… = (ab)/99

Put as many 9s in the denominator as the length of the repeating block (one digit 9, two digits 99)

An Infinite Sum, Seen Geometrically

Pieces shaded3

Series of halves

12 + 14 + 18 + ⋯ = 1/21 - 1/2 = 1

First term 1/2, ratio 1/2 — the sum is exactly 1

🟧 Half of a Half of a Half…

①Shading half the square gives 1/2

②Shading half of what remains adds 1/4

③Repeating forever, the shaded area sums to the whole square, namely 1

Sum of Areas of Similar Figures

Example 2

A square of side 2 has a new square formed by joining the midpoints of its sides, repeated forever. Find the sum of the areas of all the squares.

The midpoint square has half the area of the previous one (ratio 1/2).

The first square has area 4, so S = 4/(1−1/2).

S = 41/2 = 8

▸ 8

If the similarity ratio is k, the area ratio is k² — taking the area ratio as the common ratio is the key.

Wrap-up

Application essentials

|r|<1 ⇒ S = a1 - r

Repeating decimals, figure areas and lengths all reduce to finding first term a and ratio r

2021 provincial mock exam Math type, adapted

A square S₁ of side 6 has a square S₂ formed by joining the midpoints of its sides, repeated forever. Find the sum of the areas of all the squares.

①48

②60

③72

④96

⑤Diverges

▸ ③ 72

The midpoint square has half the area of the previous one (ratio 1/2).

Since S₁ has area 36, S = 36/(1−1/2).

S = 361/2 = 72

🎯 Exam Points

①Repeating decimals are geometric series with ratio 1/10 or 1/100

②Length of the repeating block = number of 9s in the denominator

③For figures, get the ratio from similarity: area ratio k², length ratio k

④Pin the first term as the figure’s “starting value”

⑤Always check |r|<1 then use S=a/(1−r)

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