Interesting sequences and min max concept

Question

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:
  ×   = ab = (√ab)2
Note:
AM × HM = GM2
NOT
AM × HM = GM
Very common GMAT property.

Usernamevisible · Answer

Min-Max Problems (GMAT Focus) - Master Summary

Before solving, always check:
Must values be integers?
Is zero allowed?
Can values repeat?
Are negative values allowed?
Are values required to be distinct?

These restrictions often determine the entire solution.

Four Standard Min-Max Moves
To maximize the largest value:
→ Minimize all other values.
To minimize the largest value:
→ Maximize all other values.
To maximize the smallest value:
→ Make values as equal as possible.
To minimize the smallest value:
→ Push value away from the others as much as allowed.

Always determine whether decimals/fractions are allowed.
Zero is often the smallest legal value.
If zero is allowed:
• Maximum values often increase
• Minimum values often decrease

Set-Based Min-Max
For two sets:
Maximum overlap
= Smaller set
Minimum overlap
= A + B − Total
(if positive, otherwise 0)
Formula:
A ∪ B = A + B − A∩B

Three-Set Venn Diagram Rules
Maximum triple overlap
= Smallest set
Minimum triple overlap
= A + B + C − 2(Total)
(if positive, otherwise 0)

For percentages:
Minimum triple overlap
= A + B + C − 200
(if positive, otherwise 0)

Interesting sequences and min max concept

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