Arithmetic Sequences: Finding Common Differences, Nth Terms, and Sums, Lecture notes of Business

Download Arithmetic Sequences: Finding Common Differences, Nth Terms, and Sums and more Lecture notes Business in PDF only on Docsity! Arithmetic Sequences This section will define and discuss arithmetic sequences, nth term formulas, and the sum formulas for finite arithmetic series. An arithmetic sequence may be defined as: A sequence {an} is arithmetic if each pair of consecutive terms differs by the same amount, d = ai – ai – 1. The number d is called the common difference in the sequence. (Note: When the formula given in this definition is rewritten as ai = ai – 1 + d it is known as a recursive formula because it defines a given term by referencing a proceeding term.) The following example will show how to find the common difference of an arithmetic sequence. Example 1: Find the common difference of the arithmetic sequence an = 3 – 5n. Solution: Step 1: Substitute. Substitute the example formula into the definition formula. ( ) ( ) 1 3 5 3 5 1 i id a a i i −= − ⎡ ⎤ ⎡= − − − −⎣ ⎦ ⎣ ⎤⎦ Step 2: Solve for d. ( ) ( )d i i i i 3 5 3 5 1 3 5 3 5 5 5 ⎡ ⎤ ⎡= − − − −⎣ ⎦ ⎣ = − − + − = − ⎤⎦ The common difference for this arithmetic sequence is d = –5. The nth term of an arithmetic sequence is defined as: The nth term of an arithmetic sequence, whose first term is a1 and whose common difference is d, is given by the formula an = a1 + (n – 1)d. The following examples will show how to find specific terms of an arithmetic sequence. By Ewald Fox SLAC/San Antonio College 1 Example 2: Find the first four terms and then the twentieth term of the arithmetic sequence whose first term is –1 and whose common difference is 4. Solution: Step 1: Substitute. Since it was given that a1 = –1 and d = 4, the solutions for the requested terms are found by substitution into the defined formula: ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( na a n d a GIVEN a a a a 1 1 2 3 4 20 1 1 1 4 1 3 1 4 1 4 1 4 1 20 1 4 = + − = − = − + = − + − = − + − = − + − 2 −1 ) Step 2: Solve. ( ) ( ) ( ) ( ) a a a a a 1 2 3 4 20 1 1 1 4 3 1 2 4 7 1 3 4 11 1 19 4 75 = − = − + = = − + = = − + = = − + = Example 3: Find the thirty-eighth term of the arithmetic sequence whose first term is 8 and whose nth term is given by ai+1 = ai – 7. Solution: Step 1: Solve for d. ( ) i i i i d a a a a 1 7 7 += − = − − = − By Ewald Fox SLAC/San Antonio College 2