Sums & 1 + x + x 2 + + x n G = n x - x 2 - - x n - x n + 1 - - PowerPoint PPT Presentation

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Mathematics for Computer Science MIT 6.042J/18.062J

Sums & Money

Albert R Meyer, April 10, 2013 geometric-sum.1 geometric-sum.2 Albert R Meyer, April 10, 2013

Geometric Series

G =

n

1 + x + x2 +⋯+ xn −xG = −

n

x - x2 -⋯- xn - xn+1

geometric-sum.3 Albert R Meyer, April 10, 2013

Geometric Series

G =

n

1 + x + x2 +⋯+ xn −xG = −

n

x - x2 -⋯- xn - xn+1

geometric-sum.4 Albert R Meyer, April 10, 2013

Geometric Series

G =

n

1 + x + x2 +⋯+ xn −xG = −

n

x - x2 -⋯- xn - xn+1

2

geometric-sum.5 Albert R Meyer, April 10, 2013

Geometric Series

G =

n

1 + x + x2 +⋯+ xn −xG = −

n

x - x2 -⋯- xn - xn+1

1 - xn+1

geometric-sum.6 Albert R Meyer, April 10, 2013

Geometric Series

G −

n

xG =

n

1

• xn+1

geometric-sum.7 Albert R Meyer, April 10, 2013

Geometric Series

G −

n

xG =

n

1 - xn+1

1−xn+1 G =

n

1−x

geometric-sum.8 Albert R Meyer, April 10, 2013

Geometric Series

Consider infinite sum (series)

∞ 1+ x + x2 +⋯+ xn-1 + xn +⋯= ∑xi

i=0

n+1 n

1-x G = 1-x

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geometric-sum.9 Albert R Meyer, April 10, 2013

Infinite Geometric Series

1-lim

n+1 n

x 1 limGn =

→∞

=

n→∞

1-x 1-x

1-xn+1 Gn = 1-x

geometric-sum.10 Albert R Meyer, April 10, 2013

Infinite Geometric Series

for |x| < 1

∞

∑

i

1 x =

i=0

1-x

geometric-sum.11 Albert R Meyer, April 10, 2013

The future value of $$

I will pay you $100 in 1 year, if you will pay me $X now.

geometric-sum.12 Albert R Meyer, April 10, 2013

The future value of $$ My bank will pay me 3% interest. define bankrate b ::= 1.03

bank increases my $$ by this factor in 1 year.

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geometric-sum.13 Albert R Meyer, April 10, 2013

If I deposit your $X now, I will have $b X in 1 year. So I wont lose money as long as

b

The future value of $$

X ≥ 100 X ≥ $100/1.03 ≈ $97.09

geometric-sum.14 Albert R Meyer, April 10, 2013

The future value of $$

• $1 in 1 year is worth $0.9709 now.
• $r last year is worth $1 today,

where r ::= 1/b.

• So $n paid in 2 years is worth

$nr paid in 1 year, and is worth $nr2 today.

geometric-sum.15 Albert R Meyer, April 10, 2013

The future value of $$

$n paid k years from now is worth $n·rk today where r ::= 1/bankrate.

geometric-sum.16 Albert R Meyer, April 10, 2013

Annuities

I pay you $100/year for 10 years, if you will pay me $Y now. I cant lose if you pay me 100r + 100r2 + 100r3 + + 100r10 = 100r(1+ r + + r9) = 100r(1r10)/(1r) = $853.02

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geometric-sum.17 Albert R Meyer, April 10, 2013

Annuities

I pay you $100/year for 10 years, if you will pay me $853.02.

QUICKIE: If bankrates unexpectedly increase in the next few years,

• A. You come out ahead
• B. The deal stays fair
• C. I come out ahead

geometric-sum.18 Albert R Meyer, April 10, 2013

Manipulating Sums

           

n

∑

n+1 i

d d 1-x x = dx

i=0

dx 1-x

n n n+1 i-1 i

1 d 1-x   ix = ix =  

i=0 i=1

x dx 1-x  

∑ ∑

geometric-sum.19 Albert R Meyer, April 10, 2013

Manipulating Sums

n n+1 n+2 i-1

x-(n+1)x +nx ix =

2 i=1

∑

(1-x)

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