Chapter 5 - Sequences and series
For how long must Magnus invest $4000 at 6.45% compounded half-yearly, for it to amount to $10000?
Use calculator: Step 1: Press APPS, select 1: Finance, then select 1: TVM Solver Step 2: Fill in what you are given N = ? I% = 6.45 PV = -4000 PMT = 0 FV = 10000 P/Y = 2 C/Y = 2 Step 3: Move the cursor to N, press ALPHA ENTER. So, 29 half-years are required, which is 14.5 years.
Sally invests $15000 in an account that pays 4.25% p.a. compounded monthly. How much is her investment worth after 5 years?
Use calculator: Step 1: Press APPS, select 1: Finance, then select 1: TVM Solver Step 2: Fill in what you are given N = 5 x 12 I% = 4.25 PV = 15000 PMT = 0 FV = ? P/Y = 12 C/Y = 12 Step 3: Move the cursor to FV, press ALPHA ENTER to display the future value of the investment. So, the value of the investment after 5 years is ≈ $18544.53
Find the general term Uₙ for an arithmetic sequence with U₃ = 8 and U₈ = -17.
U₃ = 8 U₈ = -17 ∴ U₁ + 2d = 8 ∴ U₁ + 7d = -17 Then solve simultaneously: -U₁ - 2d = -8 U₁ + 7d = -17 5d = -25 d = -5 Find U₁: U₁ + 2(-5) = 8 U₁ - 10 = 8 U₁ = 18 Now plug numbers into general term formula: Uₙ = U₁ + (n - 1)d Uₙ = 18 - 5(n - 1) Uₙ = 18 - 5n + 5 Uₙ = 23 - 5n
depreciation formula
Uₙ = U₀ (i - d)ⁿ U₀ - initial value of item d - rate of depreciation per annum
exponential decay
Uₙ = U₀ (i - r)ⁿ
exponential growth
Uₙ = U₀ x rⁿ or Uₙ = U₀ (i + r)ⁿ
Sam wants to buy a scooter that costs $1500. He deposits $1000 in a bank that pays 7.5% interest compounded annually. How long will it take before he can buy the scooter?
Uₙ = U₀ x rⁿ 1500 = 1000(1.075)ⁿ n = 5.606 Use calculator: Step 1: MATH → B: SOLVER 1000(1.075)ⁿ - 1500 = 0 Step 2: press ALPHA enter So, it will take about 6 years.
Say you invest $1650 in a bank that pays 7.5% interest compounded annually. How much money will you have after 11 years?
Uₙ = U₀ x rⁿ U₁₁ = 1650(1.075)¹¹ U₁₁ = $3655
general term formula (arithmetic)
Uₙ = U₁ + (n - 1)d
general term formula (geometric)
Uₙ = U₁rⁿ⁻¹
sigma notation
We can write → U₁ + U₂ + U₃ + U₄ + ... + Uₙ as seen in the image provided. It would read as "the sum of all numbers of the form Uk where k = 1, 2, 3, ..., up to n".
arithmetic sequence
a sequence in which each term differs from the previous one by the same fixed number (we refer to this number as the common difference d).
An industrial dishwasher was purchased for £2400 and depreciated by 15% each year. a) Find its value after 6 years. b) By how much did it depreciate?
a) U₆ = U₀ x (1 - d)⁶ = 2400 x (0.85)⁶ {since 15% = 0.15) ≈ £905.16 b) The depreciation = £2400 - £905.16 = £1494.84
Ernie invested $5000 in an account for 3 years at 3.6% p.a. interest compounded quarterly. Inflation over the period averaged 2% per year. a) Calculate the value of the investment after 3 years. b) Find the real value of the investment by indexing it for inflation.
a) Uₙ = U₀ (i + r)ⁿ U₁₂ = 5000 (1 + 0.036/4)³*⁴ U₁₂ = $5567.55 b) (real value)(1 + rate)ⁿ = future value (real value)(1.02)³ = 5567.55 real value = $5246.43
Consider the sequence 1, 4, 9, 16, 25, ... a) Write down an expression for Sₙ. b) Find Sₙ for n = 1, 2, 3, 4, and 5.
a) Sₙ = 1² + 2² + 3² + 4² + ... + n² b) S₁ = 1 S₂ = 1 + 4 = 5 S₃ = 1 + 4 + 9 = 14 S₄ = 1 + 4 + 9 + 16 = 30 S₅ = 1 + 4 + 9 + 16 + 25 = 55
The inflation rate of a country was calculated at 4.48% per annum. a) Given that the same rate continues for the next five years, work out the percentage increase due to inflation at the end of the five years. b) A computer game costs $35 today. Calculate what you would expect it to cost next year due to an inflation adjustment.
a) Uₙ = U₀ x rⁿ r = 1 + 0.0448 ; say U₀ = 1 U₅ = 1(1.0448)⁵ U₅ = 1.245 → increase of 24.5% b) 35 x 1.0448 = $36.57
Gemma invested $4000 in an account for 5 years at 4.8% p.a. interest compounded half-yearly. Inflation over the period averaged 3% per year. a) Calculate the value of the investment after 5 years. b) Find the real value of the investment by indexing it for inflation.
a) There are n = 5 x 2 = 10 time periods. Each period, the investment increases by i = 4.8% / 2 = 2.4% ∴ the amount after 5 years is U₁₀ = U₀ x (1 + i)¹⁰ = 4000 x (1.024)¹⁰ ≈ $5070.60 b) real value x (1.03)⁵ = $5070.60 real value = $5070.60 / (1.03)⁵ real value = $4373.94
real value
an investment's final value in terms of today's purchasing power
number sequence
an ordered list of numbers defined by a rule
finding the real value
divide the future value by the inflation multiplier
geometric sequence
is a sequence in which each term can be obtained from the previous one by multiplying by the same non-zero number (common ratio, r)
Find k given that 3k + 1, k, and -3 are consecutive terms of an arithmetic sequence.
k - (3k + 1) = -3 - k {equating the differences} k - 3k - 1 = -3 - k -2k - 1 = -3 - k -1 + 3 = -k + 2k k = 2
inflation
measures the rate that prices for goods increase overtime and, as a result, how much less your money can buy
inflation adjustment
the change in the price of an article that is a direct result of inflation
compound interest
the interest generated in one period will itself earn more interest in the next period
depreciation
the loss in value of an item over time
series
the sum of the terms of a sequence
A geometric sequence has U₂ = -6 and U₅ = 162. Find its general term.
(U₁r⁴) / U₁r = 162 / -6 r³ = -27 r = -3 U₁(-3) = -6 U₁ = 2 Thus Uₙ = 2 x (-3)ⁿ⁻¹
solving for common ratio
(Uₙ + 1) / Uₙ = r
b) Expand and evaluate
(½) + (¼) + (1/8) + (1/16) + (1/32) = 31/32
a) Expand and evaluate
2 + 3 + 4 + 5 + 6 + 7 + 8 = 35
Find the amount to which $1500 will grow if compounded daily at 6.75% interest for 10 years.
A = P(1 + i)ⁿ A = 1500(1 + 0.0675/365)¹⁰*³⁶⁵ A = 2845.87 *about $15 more than compounding $1500 quarterly at 6.75% interest.
Find the amount to which $1500 will grow if compounded quarterly at 6.75% interest for 10 years.
A = P(1 + i)ⁿ A = 1500(1 + 0.0675/4)¹⁰*⁴ A = 2929.50
general formula (compound interest)
A = P(1 + r/m)^mt A = P(1 + i)ⁿ similar to Uₙ = U₀ (i + r)ⁿ where → i = (r/m) and n = mt
k - 1, 2k, and 21 - k are consecutive terms of a geometric sequence. Find k.
Since the terms are geometric: (2k) / (k - 1) = (21 - k) / (2k) 4k² = 21k - 21 - k² + k 5k² - 22k + 21 = 0 (5k - 7)(k - 3) = 0 k = 7/5 or 3 To check: If k = 3 the terms are: 2, 6, 18 {r = 3}
Sigma notation on caluclator
Step 1: MATH, then select 0: summation ∑( Step 2: Fill in what you know Step 3: press ENTER
Example of compound interest
Suppose a principal of $1.00 was invested in an account paying 6% annual interest compounded monthly. How much would be in the account after one year? 1. amount after one month: 1 + (0.06/12)(1) = 1(1 + 0.005) = 1.005 2. amount after two months: 1.005(1 + 0.06/12) = 1.005(1.005) = 1.005² 3. amount after three months: 1.005²(1 + 0.06/12) = 1.005²(1.005) = 1.005³
Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence
Suppose the common difference is d. ∴ the numbers are 3, 3 + d, 3 + 2d, 3 + 3d, 3 + 4d, and 12 3 + 5d = 12 5d = 9 d = 9/5 = 1.8 So, the sequence would be 3, 4.8, 6.6, 8.4, 10.2, 12.
using technology for financial models
The TVM Solver can be used to find any variable if all the other variables are given. N - the number of compounding periods I% - interest rate per year PV - present value of the investment PMT - payment each time period FV - future value P/Y - number of payments per year C/Y - number of compounding periods per year
Find the first term of the sequence 6, 6√2, 12, 12√2 ... which exceeds 1400.
The sequence is geometric with U₁ = 6 and r = √2 So Uₙ = 6 x (√2)ⁿ⁻¹ Using calculator: Step 1: Press Y=, then enter the equation into Y₁ Step 2: Press 2ND GRAPH to view the table of values and then scroll down to the first term that exceeds 1400. ∴ The first term to exceed 1400 is U₁₇ = 1536
The initial population of rabbits on a farm was 50. The population increased by 7% each week. a. How many rabbits were present after: i) 15 weeks ii) 30 weeks b. How long will it take for the population to reach 500?
There is a fixed percentage increase each week, so the population forms a geometric sequence. U₀ = 50 and r = 1.07 ∴ the population after n weeks is → Uₙ = 50 x 1.07ⁿ a. i) U₁₅ = 50 x (1.07)¹⁵ ≈ 137.95 There were 138 rabbits. ii) U₃₀ = 50 x (1.07)³⁰ ≈ 380.61 There were 381 rabbits. b. We need to find when 50 x 1.07ⁿ = 500 Use calculator: Step 1: Press MATH, select option B: SOLVER Step 2: Enter equation into equation solver eqn: 0 = 50 x 1.07ⁿ - 500 Step 3: Press ALPHA enter to get answer So, it will take approx 34.0 weeks.
Iman deposits $5000 in an account that compounds interest monthly. 2.5 years later, the account has balance $6000. What annual rate of interest has been paid?
Use calculator: Step 1: Press APPS, select 1: Finance, then select 1: TVM Solver Step 2: Fill in what you are given N = 12 x 2.5 I% = ? PV = -5000 PMT = 0 FV = 6000 P/Y = 12 C/Y = 12 Step 3: Move the cursor to I%, press ALPHA ENTER to display the future value of the investment. So, the interest rate is 7.32% p.a.
