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Learn how to solve financial problems using geometric sequences with engaging examples and easy-to-follow explanations. Explore compound interest, common ratios, and nth term calculations in this comprehensive guide.
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Supplemental Material Geometric Sequences
Motivating Examples • Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its 12th month? How much profit will it earn in its first year? • An interest-free loan of $12,000 requires monthly payments of 15% of the unpaid balance. What is the unpaid or outstanding balance after 18 payments? Problems like these can be solved using geometric sequences.
Introductory Example • Find how much $5,000 grows to when it is invested at 8% annual compound interest. Beginning Balance = $5,000 Balance at end of year 1 = $5,000 x 1.08 = $5,400 Balance at end of year 2 = $5,400 x 1.08 = $5,832 Balance at end of year 3 = $5,832 x 1.08 = $6,298.56 Balance at end of year 4 = $6,298.56 x 1.08 = $6,802.44 Balance at end of year 5 = $6,802.44 x 1.08 = $7,346.64
Introductory Example Continued The sequence of numbers $5,000, $5,400, $5,832, $6,298.56, $6,802.44, $7,346.64 is called a geometric sequence. Each number, or term, is 1.08 times the previous term. The compounding of interest is essentially a geometric sequence. A geometric sequence is a sequence of numbers, called terms, such that each term is a constant multiple of the proceeding term. Any two consecutive numbers in the sequence are separated by a constant multiple called a fixed common ratio. The common ratio is equal to r. (r= 1.08 in the above sequence since $5,400/$5,000 = 1.08)
Geometric Sequence Examples • Find the common ratio in the sequence 1, 3, 9, . . . r = 3/1 = 3 • Find the next 3 terms of the sequence 1, 3, 9, . . . 27, 81, 243 • Find the common ratio in the sequence 4, 2, 1, . . . r = 2/4 = ½ • Find the next 3 terms of the sequence 4, 2, 1, . . . 1/2, 1/4, 1/8
Generalized Way to Write a Geometric Sequence • A geometric sequence can be written as a0, a1, a2, a3, a4, + . . . a0, a0r, a0r2, a0r3, a0r4, . . . a0 = first term in the sequence a1 = a0r = second term in the sequence a2 = a0r2 = third term in the sequence a3 = a0r3 = fourth term in the sequence • The n+1st term of a geometric sequence is an = a0 rn a0 is the first term in the sequence r is the common ratio (r = a1/a0)
Examples Finding the nth Term • Find the nth term of the geometric sequence 1, 4, 16, 64, . . . an = a0 rn a0 = 1 and r = 4/1 = 4 an = a0 rn an = 1 x 4 n = 4 n • Find the 6th term of the geometric sequence 1, 4, 16, 64, . . . The 6th term of the sequence is a5. an = 4 n a5 = 4 5 = 1,024
Geometric Sequences and Compound Interest • The compounding of interest is essentially a geometric sequence. geometric sequence compound interest formula an = a0 rn FV = PV(1 + i)n common ratio = r = 1 + i initial value = a0 = PV • Consider again the sequence of numbers $5,000, $5,400, $5,832, $6,298.56, $6,802.44, $7,346.64 that comes from investing $5,000 at 8% annually compounded interest. an = $5,000 (1.08)n and FV = $5,000 (1 + .08)n
Discrete and Continuous Functions • A geometric sequence is a discrete set of points. This discrete set of points forms a continuous exponential function if you connect the points. This exponential function can be written as y = a bx where a = initial value and b = common ratio. • Continuous compounding exhibits exponential growth. an y = 5,000(1.08)x n
Application Example 1 • Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its 12th month? • Solution Month 1 : $1,000 Month 2 : $1,000 + (0.1)( $1,000) = $1,000(1.1) Month 3: $1,000(1.1)2 Find the 12th term (a11) in the geometric sequence a0 = $1,000 r = 1.1 (this comes from 100% + 10% or 1 + i) an = a0 rn a11 = $1,000 (1.1)(11) = $2,853.12
Graph for Application Example 1 an = a0 rn a0 = $1,000 and r = 1.1 an = $1,000 (1.1)n an y = $1,000(1.1)x n
Application Example 2 • If changing market conditions cause a company earning $8,000,000 in 2010 to project a loss of 2% of its profit in each of the next 4 years, what profit does it project for 2014? • Solution 2010: $8,000,000 2011 : $8,000,000 -0.02($8,000,000) = $8,000,000(0.98) 2012: $8,000,000(0.98)2 Find the 5th term (a4) in the geometric sequence a0 = $8,000,000 r = 0.98 an = a0 rn a4 = $8,000,000 (0.98)4 = $7,378,945.28
Application Example 3 • An interest-free loan of $12,000 requires monthly payments of 10% of the outstanding balance. What is the outstanding balance after 18 payments? • Solution When 10% of the outstanding balance is paid each month, 90% of the balance is left. We will assume the payment is made at the end of the month. Balance at the beginning of the first month : $12,000 Balance at the beginning of the second month: $12,000(0.9) Balance at the beginning of the third month: $12,000(0.9)2 We need to find the balance at the beginning of the 19th month an = a0 rn a18 = $12,000 (0.9)18 = $1,801.14
Graph for Application Example 3 Unpaid Balance This graph shows the unpaid balance and when the loan will be paid off. y = $12,000(0.9)x Months
Sum of the First n Terms of a Geometric Sequence • The first n terms of a geometric sequence can be written from a0 to an-1 as a0, a0r, a0r2, a0r3 , . . . , a0 rn-1 • We let Sn represent the sum of the first n terms on the sequence (1) Sn = a0 + a0r + a0r2 + a0r3 + . . . + a0 rn-1 • If we multiply equation (1) by r, we have (2) rSn = a0r + a0r2 + a0r3 + a0r4 + . . . + a0 rn • Subtracting equation (2) form equation (1), we obtain Sn – rSn = a0 + (a0r – a0r) + (a0r2 – a0r2 ) + (a0 rn-1- a0 rn-1) – a0 rn • Thus Sn (1 – r) = a0 – a0 rn Sn = a0 (1 - rn)/(1 – r)
Sum of the First n Terms of a Geometric Sequence • The sum of the first n terms of a geometric sequence with first term a0 and common ratio r is • This formula only works if r is not equal to 1.
Application Example 4 • Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its first year? • Solution The profit it will earn in its first year is the sum of the profit in the first twelve months. Sn = a0 (1 – rn)/(1 – r) n = 12 a0 = $1,000 r = 1 + i = 1.1 S12 = $1,000(1 - (1.1)12)/(1 – 1.1) S12 = $21,384.28 NOTE: You do include the first term when finding the sum so, you are summing up 12 numbers!
