Download NOTES: Implicit vs. Explicit Formulas and more Study notes Analytical Geometry and Calculus in PDF only on Docsity! NOTES: Implicit vs. Explicit Formulas CONSIDER THIS: A pyramid of logs has 2 logs in the top row, 4 logs in the second row from the top, 6 logs in the third row from the top, and so on, until there are 200 logs in the bottom row. ๏ฝ Write and interpret the first 10 terms of the sequence of numbers generated from the example. ๏ฝ Identify the pattern in the sequence of numbers. ๏ฝ Write the formula for the nth term of the sequence and use it to find the number of logs in the 76th row. ๏ฝ Compute the number of logs in the first 12 rows combined. ๏ฝ What is the total number of logs in the pyramid? CONSIDER THE SEQUENCE: 1, 4, 7, 10, 13,โฆ ๏ฝ ๐ = the term number (think of it as the termโs place in line) ๏ฝ ๐๐ = the nth term ๏ซ ๐1 represents the FIRST term. ๏ซ ๐2 represents the SECOND term. ๏ซ ๐3 represents the THIRD term. ๏ซ ๐4 represents the FOURTH term, etcโฆ ๏ฝ ๐๐โ1 = the previous term ๏ฝ ๐๐+1 = the next term ๏ฝ IMPLICIT FORMULA: requires knowing the previous term ๏ฝ EXPLICIT FORMULA: requires only knowing the desired n. FILL IN THE CHART: Find the first six terms for each sequence: ๏ฝ ๐๐ = 3๐ ๏ฝ ๐๐ = ๐ + 5 ๏ฝ ๐๐ = ๐2 โ 3๐ ๏ฝ ๐๐ = 2๐๐โ1, ๐1 = 5 ๏ฝ ๐๐ = (๐๐โ1) 2, ๐1 = 2 ๏ฝ ๐๐ = ๐๐โ1 + 4, ๐1 = 3 NOTES: ARITHMETIC SEQUENCES ARITHMETIC, GEOMETRIC, or NEITHER? ๏ฝ An arithmetic sequence is one where a constant value is _______________ to each term to get the next term. example: {5, 7, 9, 11, โฆ} ๏ฝ A geometric sequence is one where a constant value is _______________by each term to get the next term. example: {5, 10, 20, 40, โฆ} ๏ฝ EXAMPLE: Determine whether each of the following sequences is arithmetic, geometric, or neither: a. { 1 2 , 1 4 , 1 8 , 1 16 , . . . } b. {9, -1, -11, -21, ...} c. {0, 1, 1, 2, 3, 5, 8, 13, 21,...} ARITHMETIC SEQUENCES: ๏ฝ A sequence is arithmetic if there exists a number d, called the common ________________________, such that for ๐ = ๐๐ โ ๐๐โ1 for ๐ โฅ 2. ๏ฝ In other words, if we start with a particular first term, and then add the same number successively, we obtain an arithmetic sequence. ๏ฝ Example: Write an explicit formula for the sequence {10, 15, 20, 25, โฆ}. In general, the explicit formula for an arithmetic sequence is given by _________________________________________.
