Question

1. numbers the multiples of two form a sequence as follows: 2, 4, 6, 8, 10, 12, 14, 16, … describe the sequence you see. what about the multiples of three? four? five? 2. olympics the summer olympics occur every four years. if the last summer olympics happened in 2008, when are the next three times that it will occur? describe the sequence the olympic years form. 3. baby - sitting tonya charges $3.50 per hour to baby - sit. the sequence $3.50, $7.00, $10.50, $14.00, … represents how much she charges for each subsequent hour. for example, $10.50 is the third term that represents how much she charges for 3 hours. what are the next three terms in the sequence? how much does she charge for 7 hours of baby - sitting? 4. jogging luther starts jogging 8 minutes on the first day and then increases his time by 4 minutes each day. how many minutes will he jog the fifth day?

Explanation:

Problem 1: Numbers (Multiples of 2 Sequence)

Step1: Identify the pattern

The sequence is \(2, 4, 6, 8, 10, 12, 14, 16, \dots\). This is an arithmetic sequence where each term is obtained by adding \(2\) to the previous term (since it's multiples of \(2\): \(2\times1 = 2\), \(2\times2 = 4\), \(2\times3 = 6\), etc.).

Step2: Find the nth term formula

The general formula for the \(n\)-th term of this sequence (multiples of \(2\)) is \(a_n=2n\), where \(n\) is the term number (\(n = 1, 2, 3, \dots\)).

Step3: Calculate terms for \(n = 3, 4, 5\)

• For \(n = 3\): \(a_3=2\times3 = 6\)
• For \(n = 4\): \(a_4=2\times4 = 8\)
• For \(n = 5\): \(a_5=2\times5 = 10\)

Step1: Identify the pattern

The Summer Olympics occur every \(4\) years. If the last one was in \(2008\), the sequence of years is an arithmetic sequence with first term \(a_1 = 2008\) and common difference \(d = 4\).

Step2: Find the nth term formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Here, \(a_1 = 2008\), \(d = 4\), so \(a_n=2008+4(n - 1)=2004 + 4n\).

Step3: Calculate next three terms (\(n = 2, 3, 4\) for next three Olympics after \(2008\))

• For \(n = 2\) (next Olympics): \(a_2=2008 + 4(1)=2012\)
• For \(n = 3\): \(a_3=2008 + 4(2)=2016\)
• For \(n = 4\): \(a_4=2008 + 4(3)=2020\)

Step1: Identify the pattern

Tonya charges \(\$3.50\) per hour. The sequence of costs is \(\$3.50\), \(\$7.00\), \(\$10.50\), \(\$14.00\), \(\dots\). This is an arithmetic sequence with first term \(a_1 = 3.50\) and common difference \(d = 3.50\) (since each hour adds \(\$3.50\)). The formula for the \(n\)-th term is \(a_n=a_1+(n - 1)d=3.50n\) (since \(d = a_1\), it's also a geometric sequence with ratio \(2\), but arithmetic is simpler here as it's linear per hour).

Step2: Cost for 3 hours

For \(n = 3\) (3 hours), \(a_3=3.50\times3=\$10.50\) (which matches the example).

Step3: Next three terms (for \(n = 4, 5, 6\))

• \(n = 4\): \(a_4=3.50\times4=\$14.00\) (given), \(n = 5\): \(a_5=3.50\times5=\$17.50\)
• \(n = 6\): \(a_6=3.50\times6=\$21.00\)
• \(n = 7\): \(a_7=3.50\times7=\$24.50\)

Answer:

• Multiple of 3: \(6\) (since \(2\times3 = 6\))
• Multiple of 4: \(8\) (since \(2\times4 = 8\))
• Multiple of 5: \(10\) (since \(2\times5 = 10\))

Problem 2: Olympics (Sequence of Olympic Years)