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9.1 – Geometric Sequences

Key Terms

  • Arithmetic Sequences – Sequences in which the differences between consecutive terms are constant.
  • Explicit Formula – A formula that defines the nth term of a sequence using only the index, n.
    • It is a rule for a sequence with n as the term number.
  • Geometric Sequence – A sequence of numbers in which the ratio of any two successive terms is equal.
  • Geometric Series – The sum of the terms in a sequence of numbers in which there is a common ratio between any two consecutive terms. (In other words, it is the sum of a geometric sequence.)
  • Partial Sum – The sum of the first n terms of a sequence. Denoted by the symbol S_n.
  • Recursive Formula – A rule that defines the nth term of a sequence or series in terms of one or more previous terms.

Notes

Arithmetic Sequences
  • Linear functions change at a constant rate.
  • Arithmetic sequences change at a constant rate.
  • Arithmetic sequences are linear functions.
  • Follows the same pattern as a linear formula: y = mx + b
  • Graph: straight line
  • Explicit Formula for Arithmetic Sequences
    • An explicit formula describes the relationship of a number to its place in the sequence.
    • These work for very simple formulas that involve skip counting (common multiples).
    • a_n=xn

  • Ex. If each nth term is 5 more than the previous term, you can use 5n to represent this
    • a_n=5n
    • Because the sequence is 5, 10, 15, 20…5n
      • The 1st term is 5
      • The 2nd term is 10
      • The 3rd term is 15
      • The 4th term is 20
      • The nth term is 5 times n.
  • Arithmetic Sequence Formulaa_n=a_1+(n-1)\bullet d
    • Some sequences use common multiples, but are shifted up or down.
      • Ex. for instance, 4, 9, 14, 19… is still counting up by 5, but it’s not as common to start with 4.
      • You will need to use a formula for an Arithmetic Sequence instead of the explicit formula.
        • a_n=a_1+(n-1)\bullet d
        • 1st term: a_1=4+(1-1)\bullet 5=4
        • 2nd term: a_2=4+(2-1)\bullet 5=9
        • 3rd term: a_3=4+(3-1)\bullet 5=14
        • And so on…

 

Recursive Formula
  • A formula used to determine the next term of a sequence using one or more of the preceding terms.
  • Ask yourself:
    • What do I know? (terms of a sequence)
    • What is being added to get the next number? (Find the constant value.)

Alg2B 9.1 Recursive

 

Geometric Sequences
  • Like the terms in an arithmetic sequence, the terms in a geometric sequence are connected by a common number, but they are linked by multiplication or division instead of by addition or subtraction.
  • A geometric sequence changes by a common ratio (a fraction).
    • In a geometric sequence, the ratio between consecutive terms is constant.
  • To find the ratio, divide the 2nd term by the 1st term.
    • Ex. 2, 6, 18, 54, 162…
      • Divide 6 by 2, and you find the common ratio: 3
    • Ex. 1000, 100, 10, 1, 1/10, 1/100…
      • Divide 100 by 1000, and you find the common ratio: 1/10
  • Graph: Exponential, curved
    • If the common ratio of a geometric sequence is greater than 1, the graph of the geometric sequence will be similar to that of a function that shows exponential growth.
    • If the common ratio is less than 1, the graph will be similar to that of a function that shows exponential decay.
  • Geometric Sequence Explicit Formula
    • a_n=a_1 r^{n-1}
  • To use this formula, you need to know the 1st term and the common ratio.

Alg2B - 9.1 Explicit


  • Formula: Sigma Notation: S_n=\sum\limits_{k=1}^n a_1 r^{n-1}
    • S: sum of all values in the series (summation).
    • n: number of values in the series.
    • k: 1st value of the sequence (index of summation).
    • a: 1st term in the series.
    • r: common ratio of the geometric series.

  • Formula for Appreciation / Increase
    • a_n=a_1\bullet (1+r)^{n-1}
  • Formula for Depreciation / Decrease
    • a_n=a_1\bullet (1-r)^{n-1}

 

  • Ex. General Explicit Formula for Geometric Sequences using the sequence 3, 6, 12, 24, 48
    • We know the 1st term: 3
    • We know the ratio: 2 (because 6/3 is 2, 12/6 is 2, and so on).
      • 1st Term: a_1=3\bullet 2^{1-1}=3
      • 2nd Term: a_2=3\bullet 2^{2-1}=6
      • 3rd Term: a_3=3\bullet 2^{3-1}=12
      • 4th Term: a_4=3\bullet 2^{4-1}=48

 

Partial Sums of a Geometric Sequence
  • Geometric series are infinite, but finding the partial sums are useful in the real world.
    • Ex. finding the total owed for a mortgage or the total of a savings account that earns regular interest
  • The sum of a finite set of terms in a series is called a partial sum.
  • Formula: Partial Sums
  • Used for the Sum of the First n Terms of a Geometric Sequence
    • s_n=\frac{a_1(1-r^{n})}{1-r}
    • To use this, you need to know two things:
      • The common ratio
      • The number of terms being added
    • You can also use sigma notation
      • Example:

Alg2B - 9.1 Partial Sums

  • Real World Example
    • Partial Sums are important for saving money for future retirement use or even paying your mortgage on a home.
    • The given formula can be used to find the monthly mortgage payment based on the amount of the loan, the rate, and the time. This monthly amount can be used to find the total amount paid.
    • When you buy a house, you start out owing a lot of money to a bank (lender). You pay a lot of interest, because the amount you owe is so large.
    • As you pay down the mortgage, the total interest that you’re paying decreases because the interest is based on the amount you owe.
      • Formula used to calculate your monthly mortgage payment (including interest):
      • M=\frac{P(i(1+i)^{nt})}{(1+i)^{nt}-1} where i=\frac{r}{n}
    • Let’s say you move to a state where you can buy a house for $125,000.
      • P: principal = 125000
      • r: interest rate = 3% or 0.03
      • n: compounded number = monthly (12)
      • t: years = 20
    • First, calculate i=\frac{r}{n}: i=\frac{0.03}{12}=0.0025
    • M=\frac{125000(0.0025(1+0.0025)^{12\bullet 20})}{(1+0.0025)^{12\bullet 20}-1}
    • M=\frac{125000(0.0025(1.0025)^{240})}{(1.0025)^{240}-1}
    • M=\frac{125000(0.0025(1.820754995)}{1.820754995-1}
    • M=\frac{125000(0.0025(1.820754995)}{1.820754995-1}
    • M=\frac{125000(0.004552)}{0.820754995}
    • M=\frac{568.9859}{0.820754995}
    • Answer: M=693.25

Examples

  • Ex 1. If you were 65 and retired today, you would need about $1000 per month (again, in today’s money); but, with inflation, you will need more when you actually retire in about 50 years from now. Things will cost more in the future.
    • Worst-case scenario, the inflation rate of 0.01 will need to be added to your future income. In other words, you will need an additional 1% of $1000 each month to make up for prices going up.
      • If you plan to be in retirement for about 30 years, how much do you need to have in savings when you retire?
      • Because this is only a partial sum (using inflation, a curved increase), we’ll need the partial sum formula.
        • s_n=\frac{a_1(1-r^{n})}{1-r}
        • r: rate (0.01 + the whole amount); so, (1.01)
          • a_1 is the 1st year: 1000
          • a_2 is the 2nd year: 1000(1.01)
          • a_3 is the 3rd year: 1000(1.01)^2
          • and so on…s_n=\frac{1000(1-1.01^{n})}{1-1.01}
        • Setup: s_n=\frac{1000(1-1.01^{n})}{-0.01}
        • s_n=-100000(1-1.01^{n} because 1000 divided by -0.01 equals -1000000.
        • So, for 30 years, we will have 360 months: n = 360
        • s_n=-100000(1-1.01^{360})
          • Answer: $3,495,000 is what you will need in savings to retire and live for about 30 more years.
  • Ex 2. What is the value of the fourth term in a geometric sequence for which a_1=30 and r=\frac{1}{2}?
  • Formula: a_n=a_1\bullet r^{n-1}
    • a_n=30\bullet \frac{1}{2}^{4-1}
    • a_n=30\bullet \frac{1}{2}^{3}
    • a_n=30\bullet 0.125
      • Answer: 3.75
  • Ex 3. The second term in a geometric sequence is 50. The fourth term in the same sequence is 112.5. What is the common ratio in this sequence?
  • Formula: a_n=a_1\bullet r^{n-1}
    • Setup: Since we don’t know the 1st term, let’s make 50 the 1st term for now. That makes 112.5 the 3rd term.
      • Substitute: a_3=50\bullet r^{3-1}
      • Simplify: 112.5=50\bullet r^{2}
      • Divide both sides by 50: 2.25=r^{2}
      • Take the square root of both sides: 1.5=r
  • Ex 4. What is the sum of the first five terms of a geometric series with a1 = 10 and r = 1/5? Answer using an improper fraction.
  • Partial Sum Formula: s_n=\frac{a_1(1-r^{n})}{1-r}
    • Substitute: s_5=\frac{10(1-\frac{1}{5}^{5})}{1-\frac{1}{5}}
    • Simplify: s_5=\frac{10(1-\frac{1}{5}^{5})}{1-\frac{1}{5}}
    • Simplify More: s_5=\frac{10(1-\frac{1}{3125})}{\frac{5}{5}-\frac{1}{5}}
    • s_5=\frac{10(\frac{3125}{3125}-\frac{1}{3125})}{\frac{4}{5}}
    • s_5=\frac{10(\frac{3124}{3125})}{\frac{4}{5}}
    • s_5=\frac{\frac{31240}{3125}}{\frac{4}{5}}
    • Keep Change Flip: s_5=\frac{31240}{3125}\bullet \frac{5}{4}
    • Cancel: s_5=\frac{7810}{625}
    • Divide by \frac{5}{5} to get the answer: \frac{1562}{125}
  • Ex 5. The original value of a car is $18,000, and it depreciates (loses value) by 15% each year. What is the value of the car after three years?
    • In this case, we are depreciating by 15% from the WHOLE value of the car. So, the rate is actually (1-0.15).
    • Since we want the rate AFTER 3 years, we need to use n = 4 because we want to get 3 full years of depreciation and 4 – 1 = 3.
      • a_4=18000\bullet (1-0.15)^{4-1}
      • a_4=18000\bullet (0.85)^{3}
      • a_4=18000\bullet 0.614125
      • Answer: $11,054.25
  • Ex 6. A ball is dropped from a height of 12 feet and returns to a height that is one-half of the height from which it fell.
    • The ball continues to bounce half the height of the previous bounce each time.
    • How far will the ball have traveled when it hits the ground for the fourth time?
    • Setup: Do this by drawing.  A ball drops 12 feet.  When it hits the ground, that’s the 1st time.
      • The ball goes up 1/2 way, so 6 feet, but then it comes back down.  That’s the 2nd time it hits the ground (adding 12 feet from the 6 up and 6 back down).
      • The ball goes up 1/2 way again, so 3 feet, then it comes back down.  That’s the 3rd time it hits the ground (adding 6 feet from the 3 up and 3 back down).
      • The ball goes up 1/2 way again, so 1.5 feet, then it comes back down.  That’s the 4th time it hits the ground (adding 3 feet from the 1.5 up and 1.5 back down).
      • Add up all the distance: 12 + 12 + 6 + 3.
        • Answer: 33 feet
  • Ex 7. If your starting salary were $35,000 and you received a 6% increase at the end of every year for 16 years, what would be the total amount, in dollars, you would have earned over the first 16 years that you worked?
    • Setup: Use the Partial Sum Formula to add up the TOTAL amount earned over 16 years.
    • S_16=\frac{35000(1-(1.06^{17-1}))}{1-1.06}
    • S_16=\frac{35000(1-(2.54))}{-0.06}
    • S_16=\frac{-53912.31}{-0.06}
    • Answer: $898,538
  • Ex 8. It is true that in a geometric sequence where r > 1, the terms always increase.
  • Ex 9. It is true that in a geometric sequence, the term a_{n+1} can be smaller than the term a_n.

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