- AP Shortcut GMAT Loves
To find:
t10 + t11 + ... + t18
Use:
(Sum first 18) − (Sum first 9)
Very common technique.
- Consecutive Integers
Special AP:
1,2,3,4,5,...
Difference = 1
Sum of First n Integers
Must know:
1+2+3+⋯+n = n(n+1)/2
- Sum of Consecutive Even Integers
Sequence:
2,4,6,8,...
If there are n terms:
Sum = n(n+1)
- Sum of Consecutive Odd Integers
Sequence:
1,3,5,7,...
If there are n terms:
Sum = n2
Must Memorize
1 = 12
1+3 = 22
1+3+5 = 32
1+3+5+7 = 42
Very common GMAT pattern.
- Alternating Sign Sequences
Very common 700+ topic.
Example:
1/2 − 1/4 + 1/8 − 1/16 + ...
Notice:
Signs alternate
Terms shrink
GMAT Trick
Group terms.
(1/2 − 1/4) + (1/8 − 1/16) + ⋯
Every bracket is positive.
This helps estimate ranges quickly.
Whenever you see:
a−b+c−d+e−f+⋯
and
signs alternate
magnitudes get smaller
don't calculate the sum.
Instead:
Pair terms.
Create a lower bound.
Create an upper bound.
Find the range.
This is the real concept being tested, not GP formulas. The question is testing estimation and bounding of alternating sequences.
- AM, GM and HM
Three important means.
Arithmetic Mean (AM)
AM = (a+b)/2
Geometric Mean (GM)
GM = √(ab)
Harmonic Mean (HM)
HM = 2ab/(a+b)
Golden Inequality
For all positive numbers:
AM ≥ GM ≥ HM
Must memorize.
Example
a = 50
b = 2
AM:
(50+2)/2
= 26
GM:
√100
= 10
HM:
(2×50×2)/(50+2)
= 200/52
≈ 3.85
Therefore
26 > 10 > 3.85
Special Identity
For two positive numbers:
AM × HM = GM2
Equivalent form:
[(a+b)/2] × [2ab/(a+b)] = ab = (√ab)2
Note:
AM × HM = GM2
NOT
AM × HM = GM
Very common GMAT property.