  Assignment 71

 "Where there is number there is beauty." - (Proclus, 410-485) Statistics: Our word statistics comes from the Latin words stato ("state") and statista ("one who serves the state" or "politician"). Statistics were originally facts or data that pertained to the state. "He uses statistics as a drunken man uses a lamp post... for support, rather than illumination." ANDREW LANG Why was Herkimer delighted when his favorite sea gull found a mate? Answer: He believed that one good tern deserves another. Herky wants to know: If love is blind, why is lingerie so popular? If a person who plays the piano is a pianist, why isn't a person who drives a race car called a racist? ASSIGNMENT #71 Reading: Review Section 11.3, as necessary. Exercises: Page 671, 64-69. Find the sum of each geometric series (a) by using the sum formula;(b) by using SUM and SEQUENCE on your calculator. Mathematical word analysis:THOUSAND: From the Germanic roots teue and hundt. The word teue references a thickening or swelling and hundt is the basis for the modern hundred. Thousand is a swollen hundred.
Derivation of formula for sum of a geometric series (with assumption that r is not equal to 1).

(A): Sn = a + ar + ar2 + ... + arn-1 (There are n terms here.)

(B): rSn = ar + ar2 + ... + arn-1 + arn

(C) = (A)-(B): Sn - rSn = a - arn

==> Sn(1-r) = a - arn

==> Sn = (a - arn)/(1-r) = a(1 - rn)/(1-r).

Example: Sum the first six terms of the geometric series 2 + 6 + 18 + 54 + 162 + 486 + ... by (a) using the sum formula; (b) using SUM and SEQUENCE on your calculator.

(a) We have a = 2 and r = 3. Using the formula, we have S6 = 2(1-36)/(1-3) = 728.

(b) On the TI-83, sum(seq(2*3^(x-1),x,1,6,1)) = 728.

 Problem: Write the first five terms of the geometric sequence defined by an = 4(3n-1) and find the sum of the first ten terms. Solution (with communication): The first five terms are 4, 12, 36,108, and 324. Using the TI-83, the sum of the first ten terms is sum(seq(4*3^(x-1),x,1,10,1)) = 118,096. Problem: Identify a rule for the n-th term in the geometric sequence -2, 8, -32, 128, -512, ... and identify the term number for the sequence number 22463 Solution (with communication): We have a = -2 and r = -4. Hence, the n-th term is an = -2(-4)n-1. The sequence consists only of even numbers. Hence the number 22463 is not a term in the sequence.