Episode 8 — Aptitude and Reasoning / 8.17 — Geometric Progression

8.17.b Geometric Progression - Tips, Tricks and Shortcuts

Trick 1: Symmetric Selection for GP Problems

When a problem says "product of 3 numbers in GP is X":

Three terms: a/r, a, ar

Product = (a/r) * a * (ar) = a^3
So: a = cube_root(X)  (one variable eliminated immediately)

Then use other conditions (like sum) to find r.

For 4 terms: a/r^3, a/r, ar, ar^3 with common ratio r^2. For 5 terms: a/r^2, a/r, a, ar, ar^2.

Trick 2: Quick Common Ratio Identification

Divide any term by its predecessor:
r = a(n+1) / a(n)

Quick check: Verify with at least 2 pairs to confirm it is a GP.

Trap alert: If the ratio alternates (positive/negative), it could still be a GP with a negative common ratio.

Trick 3: GP or AP? Quick Distinction

AP check: a(2) - a(1) = a(3) - a(2)?  -> Constant difference
GP check: a(2)/a(1) = a(3)/a(2)?       -> Constant ratio

Three numbers a, b, c form:
  AP if: 2b = a + c
  GP if: b^2 = a * c

Trick 4: Sum to Infinity - Instant Application

For problems involving infinite GP with |r| < 1:

S(infinity) = a / (1 - r)

Common scenarios:
- Recurring decimals -> GP with r = 1/10 or 1/100
- Bouncing ball problems -> r = fraction of height
- Infinite geometric series in algebra

Quick reference for common infinite sums:

Series	r	Sum
`1 + 1/2 + 1/4 + 1/8 + ...`	1/2	2
`1 + 1/3 + 1/9 + 1/27 + ...`	1/3	3/2
`1 - 1/2 + 1/4 - 1/8 + ...`	-1/2	2/3
`1 - 1/3 + 1/9 - 1/27 + ...`	-1/3	3/4
`1/2 + 1/4 + 1/8 + ...`	1/2	1
`1/3 + 1/9 + 1/27 + ...`	1/3	1/2

Trick 5: Recurring Decimals to Fractions

0.aaa... = a/9
0.ababab... = ab/99
0.abcabcabc... = abc/999

Examples:
0.777... = 7/9
0.353535... = 35/99
0.142142142... = 142/999

For mixed recurring decimals:
0.1666... = (16 - 1)/90 = 15/90 = 1/6
0.8333... = (83 - 8)/90 = 75/90 = 5/6

General formula:

0.X(non-repeating)Y(repeating)...

Fraction = (entire number - non-repeating part) / (as many 9s as repeating digits followed by as many 0s as non-repeating digits)

Trick 6: Bouncing Ball Shortcut

A ball is dropped from height h and bounces to r times the height:

Total distance before coming to rest:
D = h * (1 + 2r) / (1 - r)

Alternative form:
D = h * (1 + r) / (1 - r)

Breakdown:
  Distance going down = h / (1 - r)
  Distance going up = hr / (1 - r)
  Total = h(1 + r) / (1 - r)

Example: Ball dropped from 100m, rebounds to 3/5 of height.

h = 100, r = 3/5

Total distance = 100 * (1 + 3/5) / (1 - 3/5)
               = 100 * (8/5) / (2/5)
               = 100 * 4
               = 400 m

Trick 7: Powers of r - Quick Mental Math

Memorize small powers for commonly tested ratios:

r	r^2	r^3	r^4	r^5	r^6
2	4	8	16	32	64
3	9	27	81	243	729
1/2	1/4	1/8	1/16	1/32	1/64
1/3	1/9	1/27	1/81	1/243	1/729
-2	4	-8	16	-32	64

Trick 8: Sum Formula Selection

Choose wisely:

If r > 1: Use S(n) = a(r^n - 1)/(r - 1)    -> avoids negatives
If r < 1: Use S(n) = a(1 - r^n)/(1 - r)    -> avoids negatives
If r = 1: S(n) = na                           -> trivial case

Trick 9: Finding nth Term from Sum

a(n) = S(n) - S(n-1)   for n >= 2
a(1) = S(1)

Quick check: If S(n) is given, always verify that a(1) = S(1) and a(2) = S(2) - S(1).

Trick 10: Geometric Mean Shortcuts

For two numbers a and b:

GM = sqrt(a * b)

Quick application: If you need to insert one GM between a and b:

The GM IS the geometric mean: sqrt(ab)

For n geometric means between a and b:

r = (b/a)^(1/(n+1))

Key relationship:

AM >= GM >= HM (for positive numbers)

AM * HM = GM^2

Trick 11: Log Trick for GP Problems

If terms are in GP, their logarithms are in AP:

GP: a, ar, ar^2, ar^3, ...
AP: log(a), log(a)+log(r), log(a)+2log(r), ...

Application: To check if large numbers form a GP, take logs and check for AP.

Trick 12: Product of GP Terms

Product of n terms of GP = (a1 * an)^(n/2)

Or equivalently for odd n:
Product = (middle term)^n

Example: Product of 2, 6, 18, 54, 162

n = 5, middle term = 18
Product = 18^5 = 1,889,568

Verify: 2 * 6 * 18 * 54 * 162 = 1,889,568 (correct)

Trick 13: Converting Sum to Infinity Problems

When S(infinity) is given, find a or r:

S = a / (1 - r)

So: a = S(1 - r)
    r = 1 - a/S = (S - a)/S

Trick 14: AGP (Arithmetico-Geometric Progression)

When each term is a product of AP and GP terms:

Sum to n terms: Use the "multiply by r and subtract" method.

S = a + (a+d)r + (a+2d)r^2 + ...
rS = ar + (a+d)r^2 + (a+2d)r^3 + ...
S - rS = a + d(r + r^2 + r^3 + ...) - last term * r

For infinite AGP (|r| < 1):
S(infinity) = a/(1-r) + dr/(1-r)^2

Trick 15: Ratio of Sums of Two GPs

If two GPs have:

Same number of terms n
Same common ratio r
First terms a1 and a2

Then:

S1/S2 = a1/a2

The ratio of sums equals the ratio of first terms.

Exam Strategy Tips

Time Management

Identify GP quickly: Look for "multiplied by," "ratio," "geometric sequence," "doubles/triples each time."
Infinite GP: If the problem mentions "sum to infinity" or "infinite series," check |r| < 1 first.
Product conditions: When product is given, always use symmetric form.

Common Traps

r can be negative: Don't assume r > 0. Check the signs of terms.
r can be a fraction: GP terms can decrease if 0 < r < 1.
Sum to infinity exists only for |r| < 1: If |r| >= 1, the infinite sum diverges.
GP terms cannot be zero: Unlike AP, no term in a GP can be zero.
Confusing AP and GP formulas: nth term of AP uses addition; nth term of GP uses multiplication/powers.

Verification Shortcuts

Check: a(2)/a(1) = a(3)/a(2) = r (common ratio is consistent)
Check: a(1) * a(3) = a(2)^2 (product of extremes = square of middle for 3 terms)
Check: Sum of infinite GP should be > a(1) if r > 0 and < a(1) if r < 0

Next: 8.17.c - Solved Examples