The Geometric Series

Disclaimer: The material in the videos below comes mainly from outside City Tech. The presentation in these videos may differ from the one given in your class. Please consult with your instructor to confirm whether a particular approach is acceptable in your class.

Geometric sequences

1. $\rhd$ Introduction to geometric sequences (10:44) Recognizing a geometric sequence, the initial term and common ratio.
2. $\rhd$ Introduction to geometric sequences (advanced) (6:38) An emphasis on the notation is given to deduce the $n^{\mathrm{th}}$ term of the sequence.
3. $\rhd$ Explicit and recursive formulas for geometric sequences (5:37) Find explicit and recursive formulas for the $n^{\mathrm{th}}$ term of the sequence $\{168, 84, 42, 21, \dots\}$.
4. * Practice: Explicit formulas for geometric series. (4 problems)

Geometric series

1. $\rhd$ Introduction to geometric series (6:16) Using a savings account as an application of geometric series.
2. $\rhd$ Geometric series (2:52) Using the sigma notation to represent the sum of the terms of the sequence.
3. $\rhd$ Finite geometric series (7:14) Deriving a formula for a finite geometric series.
4. $\rhd$ Evaluate a finite geometric series (6:54)
   1. Find the sum of the first $50$ terms in the sequence: $1+\dfrac{10}{11} + \dfrac{100}{121}+\cdots$.
   2. Evaluate $1-0.99+0.99^2-0.99^3+\cdots -0.99^{79}$.
   3. Find the sum of the first $30$ terms in the sequence given by $a_1=10$ and $a_i = a_{i-1}\dfrac{9}{10}$.
5. * Practice: Geometric series formula. (4 problems)
6. $\rhd$ Finding the sum of an infinite geometric series (19:49) Solve, for example, $8+4+2+1+\frac 1 2+\dots$, and $3+2+\frac 4 3+\frac 8 9 +\dots$, and $\sum\limits_{n=1}^\infty (\frac 4 5)^{n-1}$, and $\sum\limits_{n=1}^\infty 8(\frac 2 3)^{n-1}$
7. $\rhd$ Finding infinite geometric series (4:29) Find the sum of the geometric sequence $\frac 1 2, -\frac 1 4, \frac 1 8, \dots$.