Arithmetic and Geometric Sequences -- Sections 8.3/8.4 --

 Arithmetic Sequence: is a sequence with a common difference,  an = a1 + (n - 1)d  where d is the common difference. Geometric Sequence: is a sequence with a common ratio,  an = a1rn - 1  where r is the common ratio. Find the nth term of the following arithmetic sequence   5, 8, 11, 14, 17, ...   Then find the 100th term.
 Solve Step Spot Check d  = 8 - 5 = 3,       d  = 11 - 8 = 3  d  = 14 - 11 = 3,   d  = 17 - 14 = 3 Find the common difference d = 3 a1 =  3(1) + 2 = 5 an = a1 + (n - 1)d Use the formula for arithmetic sequence since there is a common difference a3 =  3(3) + 2 = 9 + 2 = 11 a1 = 5 and   d = 3 a1 = 5 is the first term of the sequence. a5 =  3(5) + 2 = 15 + 2 = 17 an = 5 + (n - 1)3 Plug in the values... an = 5 + 3n - 3 = 3n + 2 ... and simplify. an =  3n + 2 nth term a100 =  3(100) + 2 = 300 + 2 = 302 100th term  Find the nth term of the following arithmetic sequence   29, 25, 21, 17, 13, ...   Then find the 25th term. [Solution]  Find the sum of the first 20 terms of the arithmetic sequence 2, 6, 10,... .
 First find the nth term.  Then use the formula for the sum of an arithmetic sequence. Solve Step Spot Check d  = 6 - 2 = 4,       d  = 10 - 6 = 4 Find the common difference d = 4 a1 =  4(1) - 2 =  2 an = a1 + (n - 1)d Use the formula for arithmetic sequence. a3 =  4(3) - 2 =  12 - 2 = 10 a1 = 2 and   d = 4 a1 = 2 is the first term of the sequence. an = 2 + (n - 1)4 Plug in the values... an = 2 + 4n - 4 = 4n - 2 ... and simplify. an =  4n - 2 nth term Use the formula for the sum of an arithmetic sequence. Alternate formula n = number of terms a = first term l = last term a = 2,  l = a20 =  4(20) - 2 = 80 - 2 = 78 n = 20 Plug in the values... and simplify.  Find the sum of the first 30 terms of the arithmetic sequence 1, 5, 9,... . [Solution]  Find the nth term of the following geometric sequence   81, 27, 9, 3, 1, ...   Then find the 7th term.
 Solve Step Spot Check Find the common ratio r = 1/3 an = a1rn - 1 Use the formula for geometric sequence. a1 = 81 and   r = 1/3 a1 = 81 is the first term of the sequence. nth term 7th term  Find the nth term of the following geometric sequence  1, 5, 25, 125, 625, ...   Then find the 7th term. [Solution]  Find the value of \$10,000 left on deposit for 20 years at an annual rate of 12% compounded quarterly.
 This is a geometric sequence problem with a small change.  an = a0(1+ r)n  Let a0 = initial value. Solve Step a0 = 10,000 a0 = initial value r = 0.12 ÷ 4 = 0.03 r  = 12% annual rate divided by 4 (quarterly is 4 times a year) n = 20 × 4 = 80 n = number of years times by 4 (quarterly is 4 times a year) a80 = 10,000(1 + 0.03)80 = a80 = 10,000(1.03)80 » \$106,408.91 The account would have \$106,408.91 at the end of 20 years.  Find the value of \$20,000 left on deposit for 30 years at an annual rate of 9% compounded monthly. [Solution] Tutorials and Applets by
Joe McDonald