Sequence and Series

Question 1

Arithmetic Progression

Answer 1

y_t = a + d*t

Question 2

Geometric Progression

Answer 2

y_t = a*r^t

Question 3

Compound Interest

Answer 3

1st year: (P + rP)
2nd: P(1+r)^2
t years: P(1+r)^t

Question 4

Periodic Compounding Interest

Answer 4

P(1+r/m)^m
Where m: equal periods (e.g. half-yearly, monthly, quarterly)

Question 5

Continuous Compounding

Answer 5

limit as m approaches infinity (1+r/m)^m = e^r
1st year: Pe^r
2nd year: Pe^2r
t years: P*e^rt

Question 6

Investment Series

Answer 6

Adds $P at beginning of each year
1st year: $P
2nd: P(1+r)+P
t year: P + P(1+r) + … + P(1+r)^t-2 + P(1+r)^t-1

Question 7

Arithmetic Series Sum

Answer 7

St = [t(2a+(t-1)d)] / 2

Question 8

Partial Sum for Geometric Finite Series

Answer 8

For any r, 1-r^t if r =/ 1 and y_t = ar^t
St = a + ar + ar^2 + … + ar^t-2 + ar^t-1
St = [a(1-r^t)] / (1-r)

Question 9

Geometric series nth term

Answer 9

an = a(r)^(n-1)

Question 10

Geometric sequence nth term

Answer 10

an = r x a_(n-1)

Question 11

Geometric sequence r

Answer 11

a2/a1

Question 12

Finding sequence formula

Answer 12

1. Pattern of y0, y1, y2, y3
2. yt -> geometric series
3. flip yt to ascending order -> geometric progression with a and r
4. Sum of geometric series

Question 13

Sum to infinity for geometric series

Answer 13

when -1 < r < 1 (converges), r^t approaches 0 as t approaches infinity
Meaning St approaches a / 1-r as t -> infinity

Question 14

If r > 1 for infinite geometric series

Answer 14

Geometric series diverges and can’t use sum to infinity formula