Lake Zurich High School Arithmetic Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: April 28, 2006
Arithmetic Sequences 5, 8, 11, 14, 17, 20, … 3n+2, … Jeff Bivin -- LZHS
nth term of arithmetic sequence an = a1 + d(n – 1) Jeff Bivin -- LZHS
Find the nth term of an arithmetic sequence First term is 8 Common difference is 3 an = a1 + d(n – 1) an = 8 + 3(n – 1) an = 8 + 3n – 3 an = 3n + 5 Jeff Bivin -- LZHS
an = a1 + d(n – 1) an = -6 + 7(n – 1) an = -6 + 7n – 7 an = 7n - 13 Finding the nth term First term is -6 common difference is 7 an = a1 + d(n – 1) an = -6 + 7(n – 1) an = -6 + 7n – 7 an = 7n - 13 Jeff Bivin -- LZHS
an = a1 + d(n – 1) an = 23 + -4(n – 1) an = 23 - 4n + 4 an = -4n + 27 Finding the nth term First term is 23 common difference is -4 an = a1 + d(n – 1) an = 23 + -4(n – 1) an = 23 - 4n + 4 an = -4n + 27 Jeff Bivin -- LZHS
an = a1 + d(n – 1) a100 = 5 + 6(100 – 1) a100 = 5 + 6(99) Finding the 100th term 5, 11, 17, 23, 29, . . . an = a1 + d(n – 1) a100 = 5 + 6(100 – 1) a100 = 5 + 6(99) a100 = 5 + 594 a100 = 599 a1 = 5 d = 6 n = 100 Jeff Bivin -- LZHS
an = a1 + d(n – 1) a956 = 156 + -16(956 – 1) a956 = 156 - 16(955) Finding the 956th term a1 = 156 d = -16 n = 956 156, 140, 124, 108, . . . an = a1 + d(n – 1) a956 = 156 + -16(956 – 1) a956 = 156 - 16(955) a956 = 156 - 15280 a956 = -15124 Jeff Bivin -- LZHS
Find the Sum of the integers from 1 to 100 Jeff Bivin -- LZHS
Summing it up Sn = a1 + (a1 + d) + (a1 + 2d) + …+ an Sn = an + (an - d) + (an - 2d) + …+ a1 Jeff Bivin -- LZHS
1 + 4 + 7 + 10 + 13 + 16 + 19 a1 = 1 an = 19 n = 7 Jeff Bivin -- LZHS
4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 a1 = 4 an = 24 n = 11 Jeff Bivin -- LZHS
Find the sum of the integers from 1 to 100 a1 = 1 an = 100 n = 100 Jeff Bivin -- LZHS
Find the sum of the multiples of 3 between 9 and 1344 Sn = 9 + 12 + 15 + . . . + 1344 Jeff Bivin -- LZHS
Find the sum of the multiples of 7 between 25 and 989 Sn = 28 + 35 + 42 + . . . + 987 Jeff Bivin -- LZHS
Evaluate Sn = 16 + 19 + 22 + . . . + 82 a1 = 16 an = 82 d = 3 n = 23 Jeff Bivin -- LZHS
Evaluate Sn = -29 - 31 - 33 + . . . - 199 a1 = -29 an = -199 d = -2 Jeff Bivin -- LZHS
Find the sum of the multiples of 11 that are 4 digits in length an = 9999 d = 11 Sn = 10 01+ 1012 + 1023 + ... + 9999 Jeff Bivin -- LZHS
Review -- Arithmetic Sum of n terms nth term Jeff Bivin -- LZHS
Lake Zurich High School Geometric Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 11, 2005
Geometric Sequences 1, 2, 4, 8, 16, 32, … 2n-1, … 81, 54, 36, 24, 16, … , . . . Jeff Bivin -- LZHS
nth term of geometric sequence an = a1·r(n-1) Jeff Bivin -- LZHS
Find the nth term of the geometric sequence First term is 2 Common ratio is 3 an = a1·r(n-1) an = 2(3)(n-1) Jeff Bivin -- LZHS
Find the nth term of a geometric sequence First term is 128 Common ratio is (1/2) an = a1·r(n-1) Jeff Bivin -- LZHS
Find the nth term of the geometric sequence First term is 64 Common ratio is (3/2) an = a1·r(n-1) Jeff Bivin -- LZHS
an = a1·r(n-1) an = 3·(2)10-1 an = 3·(2)9 an = 3·(512) an = 1536 Finding the 10th term 3, 6, 12, 24, 48, . . . a1 = 3 r = 2 n = 10 an = a1·r(n-1) an = 3·(2)10-1 an = 3·(2)9 an = 3·(512) an = 1536 Jeff Bivin -- LZHS
an = a1·r(n-1) an = 2·(-5)8-1 an = 2·(-5)7 an = 2·(-78125) Finding the 8th term 2, -10, 50, -250, 1250, . . . a1 = 2 r = -5 n = 8 an = a1·r(n-1) an = 2·(-5)8-1 an = 2·(-5)7 an = 2·(-78125) an = -156250 Jeff Bivin -- LZHS
Sum it up Jeff Bivin -- LZHS
1 + 3 + 9 + 27 + 81 + 243 a1 = 1 r = 3 n = 6 Jeff Bivin -- LZHS
4 - 8 + 16 - 32 + 64 – 128 + 256 a1 = 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 = 2 + 4 + 8+…+1024 a1 = 2 r = 2 n = 10 an = 1024 Jeff Bivin -- LZHS
Evaluate = 3 + 6 + 12 +…+ 384 a1 = 3 r = 2 n = 8 an = 384 Jeff Bivin -- LZHS
an = a1·r(n-1) Review -- Geometric Sum of n terms nth term Jeff Bivin -- LZHS
Geometric Infinite Series Jeff Bivin -- LZHS
The Magic Flea (magnified for easier viewing) There is no flea like a Magic Flea Jeff Bivin -- LZHS
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
rebounds ½ of the distance from which it fell -- 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? 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS
A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft Downward = 128 + 64 + 32 + 16 + 8 + … 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS
A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft Upward = 64 + 32 + 16 + 8 + … 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS
A Bouncing Ball 128 ft 64 ft 32 ft 16 ft 8 ft Downward = 128 + 64 + 32 + 16 + 8 + … = 256 Upward = 64 + 32 + 16 + 8 + … = 128 TOTAL = 384 ft. 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS
rebounds 3/5 of the distance from which it fell -- 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? 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS
A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft Downward = 625 + 375 + 225 + 135 + 81 + … 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS
A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft Upward = 375 + 225 + 135 + 81 + … 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS
A Bouncing Ball 625 ft 375 ft 225 ft 135 ft 81 ft Downward = 625 + 375 + 225 + 135 + 81 + … = 1562.5 Upward = 375 + 225 + 135 + 81 + … = 937.5 TOTAL = 2500 ft. 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS
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 .9 = 1 That’s All Folks Jeff Bivin -- LZHS