Geometric Sequences Series By Jeffrey Bivin Lake Zurich

Geometric Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff. bivin@lz 95. org Last Updated: October 11, 2005

Geometric Sequences 1, 2, 4, 8, 16, 32, … 2 n-1, … 3, 9, 27, 81, 243, … 3 n, . . . 81, 54, 36, 24, 16, … Jeff Bivin -- LZHS , . . .

nth term of geometric sequence an = a 1 Jeff Bivin -- LZHS (n-1) ·r

Find the nth term of the geometric sequence First term is 2 Common ratio is 3 an = a 1 an = Jeff Bivin -- LZHS (n-1) ·r (n-1) 2(3)

Find the nth term of a geometric sequence First term is 128 Common ratio is (1/2) (n-1) an = a 1·r Jeff Bivin -- LZHS

Find the nth term of the geometric sequence First term is 64 Common ratio is (3/2) (n-1) an = a 1·r Jeff Bivin -- LZHS

Finding the 10 th term a 1 = 3 r=2 n = 10 Jeff Bivin -- LZHS 3, 6, 12, 24, 48, . . . (n-1) ·r an = a 1 10 -1 an = 3·(2) 9 an = 3·(2) an = 3·(512) an = 1536

Finding the 8 th term a 1 = 2 r = -5 n=8 Jeff Bivin -- LZHS 2, -10, 50, -250, 1250, . . . (n-1) ·r an = a 1 8 -1 an = 2·(-5) 7 an = 2·(-5) an = 2·(-78125) an = -156250

Sum it up Jeff Bivin -- LZHS

1 + 3 + 9 + 27 + 81 + 243 a 1 = 1 r=3 n=6 Jeff Bivin -- LZHS

4 - 8 + 16 - 32 + 64 – 128 + 256 a 1 = 4 r = -2 n=7 Jeff Bivin -- LZHS

Alternative Sum Formula We know that: Multiply by r: Simplify: Substitute: Jeff Bivin -- LZHS

Find the sum of the geometric Series Jeff Bivin -- LZHS

Evaluate a 1 = 2 r=2 n = 10 an = 1024 Jeff Bivin -- LZHS = 2 + 4 + 8+…+1024

Evaluate a 1 = 3 r=2 n=8 an = 384 Jeff Bivin -- LZHS = 3 + 6 + 12 +…+ 384

Review -- Geometric nth term an = a 1 Jeff Bivin -- LZHS (n-1) ·r Sum of n terms

Geometric Infinite Series Jeff Bivin -- LZHS

The Magic Flea (magnified for easier viewing) Jeff Bivin -- LZHS There is no flea like a Magic Flea

The Magic Flea (magnified for easier viewing) Jeff Bivin -- LZHS

Sum it up -- Infinity Jeff Bivin -- LZHS

Remember --The Magic Flea Jeff Bivin -- LZHS

Jeff Bivin -- LZHS

A Bouncing Ball rebounds ½ of the distance from which it fell -What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 128 feet tall building? Jeff Bivin -- LZHS

A Bouncing Ball Downward = 128 + 64 + 32 + 16 + 8 + … Jeff Bivin -- LZHS

A Bouncing Ball Upward Jeff Bivin -- LZHS = 64 + 32 + 16 + 8 + …

A Bouncing Ball Downward = 128 + 64 + 32 + 16 + 8 + … = 256 Upward = Jeff Bivin -- LZHS 64 + 32 + 16 + 8 + … = 128 TOTAL = 384 ft.

A Bouncing Ball rebounds 3/5 of the distance from which it fell -What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 625 feet tall building? Jeff Bivin -- LZHS

A Bouncing Ball Downward = 625 + 375 + 225 + 135 + 81 + … Jeff Bivin -- LZHS

A Bouncing Ball Upward Jeff Bivin -- LZHS = 375 + 225 + 135 + 81 + …

A Bouncing Ball Downward = 625 + 375 + 225 + 135 + 81 + … = 1562. 5 Upward = Jeff Bivin -- LZHS = 937. 5 TOTAL = 2500 ft. 375 + 225 + 135 + 81 + …

Find the sum of the series Jeff Bivin -- LZHS

Fractions - Decimals Jeff Bivin -- LZHS

Let’s try again Jeff Bivin -- LZHS

One more subtract Jeff Bivin -- LZHS

OK now a series Jeff Bivin -- LZHS

. 9 = 1 That’s All Folks Jeff Bivin -- LZHS . 9 = 1