Unit 7

Series Formulas

Geometric Series

$S_n=\frac{a(r^n-1)}{r-1}(r\ne 1)$

• a = first term
• r = common ratio (found by $r=a_{n+a}/a_n$)
• n = number of terms
• If r = 1, then $Sn=n\cdot a$

Finding r: Divide any term by the previous term. Check that ratio is consistent across all consecutive pairs.

Arithmetic Series

$S_n=\frac{n}{2}(t_1+t_n)$

Alternative form: $S_n=\frac{n}{2}[2a+(n-1)d]$

• a = first term
• d = common difference
• n = number of terms
• t₁ = first term, tₙ = last term

Kahn acc

q1

Vince went on a 3 day hiking trip. Each day, he walked 3/4 the distance that he walked the day before. He walked ‍83.25 kilometers total in the trip.

a + a * 3/4 + a * 3/4 * 3/4 = 83.25
a + 3/4a + 9/16a = 83.25

1 + 3/4 + 9/16 = 37/16

37/16a = 83.25

a = 83.25/(37/16) = 36

a = 36

q2

A new shopping mall is gaining in popularity. Every day since it opened, the number of shoppers 5% more than the number of shoppers the day before. The total number of shoppers over the first ‍  10 days is ‍ 1258.

sum = 1258

a + 1.05a + 1.05(1.05)a + … = 1258

How many shoppers were at the mall on the first day?

$$\begin{array}{rl}& Sn=a\frac{r^n-1}{r-1}\\ & \\ & 1258=a\frac{1.05^{10}-1}{1.05-1}\\ & 1258=a\frac{1.05^{10}-1}{0.05}\\ & \frac{1258}{12.57789254}=a\\ & a=100.01675527\end{array}$$