Geometric Sequences: Definition, Subsequent Terms, and nth Term, Study notes of Analytical Geometry and Calculus

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Notes: Geometric Sequences

The table shows the heights of a bungee jumper’s bounces.

The height of the bounces shown in the table above form a geometric sequence. In a geometric sequence , the ratio of successive terms is the same number r , called the common ratio.

I. What is a Geometric Sequence?

Ex 2: Find the next three terms in the geometric sequence.

Step 1 Find the value of r by dividing each term by the one before it.

The value

• of r is.

Step 2 Multiply each term by to find the next three terms.

The next three terms are

Ex 3: Find the next three terms in the geometric sequence. (^) 5, – 10, 20, –40, …

Step 1 Find the value of r by dividing each term by the one before it. 5 – 10 20 – 40

The value of r is – 2.

Step 2 Multiply each term by – 2 to find the next three terms.

The next three terms are 80, – 160, and 320.

Ex 4: Find the next three terms in the geometric sequence. (^) 512, 384, 288,…

Step 1 Find the value of r by dividing each term by the one before it.

512 384 288

The value of r is 0.75. Step 2 Multiply each term by 0.75 to find the next three terms.

The next three terms are 216, 162, and 121.5.

The pattern in the table shows that to get the n th term, multiply the first term by the common ratio raised to the power ( n – 1).

The nth term of a geometric sequence with common ration r and first term a, is

Ex 1:

The first term of a geometric sequence is 500, and the common ratio is 0.2. What is the 7th term of the sequence?

an = a 1 rn–^1 Write the formula.

a 7 = 500(0.2)^7 –^1 Substitute 500 for a 1 ,7 for n, and 0.2 for r. = 500(0.2)^6 Simplify the exponent. = 0.032 Use a calculator.

The 7th term of the sequence is 0.032.

Ex 3:

What is the 9th term of the geometric sequence 2, – 6, 18, –54, …?

2 – 6 18 – 54

an = a 1 rn–^1 Write the formula. a 9 = 2(–3)^9 –^1 Substitute 2 for a 1 ,9 for n, and

- 3 for r. = 2(–3)^8 Simplify the exponent. = 13,122 Use a calculator. The 9th term of the sequence is 13,122.

The value of r is

- 3.

When writing a function rule for a sequence with a negative common ratio, remember to enclose r in parentheses. – 212 ≠ (–2)^12

Caution

Ex 1

A ball is dropped from a tower. The table shows the heights of the ball’s bounces, which form a geometric sequence. What is the height of the 6th bounce?

Bounce Height (cm)

The value of r is 0.5.

IV. Applications

an = a 1 rn–^1 Write the formula.

a 6 = 300(0.5)^6 –^1 Substitute 300 for a 1 , 6 for n, and 0.5 for r.

= 300(0.5)^5 Simplify the exponent.

= 9.375 Use a calculator.

The height of the 6th bounce is 9.375 cm.

an = a 1 rn–^1 Write the formula.

a 10 = 10,000(0.8)^10 –^1 Substitute 10,000 for a 1 ,10 for n, and 0.8 for r.

= 10,000(0.8)^9 Simplify the exponent.

= 1342.18 Use a calculator.

In the 10th year, the car will be worth $1342.18.

Lesson Quiz: Part I

Find the next three terms in each geometric sequence.

1. 3, 15, 75, 375,…

2.

3. The first term of a geometric sequence is 300 and the common ratio is 0.6. What is the 7th term of the sequence? 4. What is the 15th term of the sequence 4, – 8, 16, –32, 64, …?