This is where the infinite geometric-series formula from earlier turns up in real situations.
A repeating decimal like 0.777, and the total area of figures that keep shrinking as you join their midpoints, are both just geometric series once you spot the first term and the ratio.
In particular, if you shade half of a square, then half of what is left, and repeat forever, the pieces fill exactly one whole square so the sum comes out to 1.
Raise the number of shaded pieces and watch the area total Sₖ inch closer and closer to 1.
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.
1
View it as an infinite geometric series with first term 7/10 and ratio 1/10.
2
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
seegongsik.com
Shading a square by halves shows 1/2 + 1/4 + 1/8 + … converges to 1
3
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.
1
The midpoint square has half the area of the previous one (ratio 1/2).
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
1
The midpoint square has half the area of the previous one (ratio 1/2).
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”