Splash Screen
Five-Minute Check (over Lesson 10–2) CCSS Then/Now New Vocabulary Key Concept: nth Term of a Geometric Sequence Example 1: Real-World Example: Find the nth Term Example 2: Write an Equation for the nth Term Example 3: Find Geometric Means Key Concept: Partial Sum of a Geometric Series Example 4: Real-World Example: Find the Sum of a Geometric Series Example 5: Sum in Sigma Notation Example 6: Find the First Term of a Series Lesson Menu
Find Sn for the arithmetic series. a1 = 4, an = 16, and n = 5 B. 20 C. 50 D. 60 5-Minute Check 1
Find Sn for the arithmetic series. a1 = 4, an = 16, and n = 5 B. 20 C. 50 D. 60 5-Minute Check 1
Find Sn for the arithmetic series. a1 = –4, an = 53, and n = 20 B. 245 C. 140 D. 70 5-Minute Check 2
Find Sn for the arithmetic series. a1 = –4, an = 53, and n = 20 B. 245 C. 140 D. 70 5-Minute Check 2
Find the first three terms of the arithmetic series for which a1 = 3, an = 33, and Sn = 108. B. 3, 9, 15 C. 3, 9, 18 D. 3, 18, 33 5-Minute Check 3
Find the first three terms of the arithmetic series for which a1 = 3, an = 33, and Sn = 108. B. 3, 9, 15 C. 3, 9, 18 D. 3, 18, 33 5-Minute Check 3
A. –1 B. 2 C. 13 D. 22 5-Minute Check 4
A. –1 B. 2 C. 13 D. 22 5-Minute Check 4
Find the sum of the first 10 positive odd integers. A. 27 B. 33 C. 69 D. 100 5-Minute Check 5
Find the sum of the first 10 positive odd integers. A. 27 B. 33 C. 69 D. 100 5-Minute Check 5
A. 2 B. 3 C. 5 D. 6 5-Minute Check 6
A. 2 B. 3 C. 5 D. 6 5-Minute Check 6
Mathematical Practices Content Standards A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Mathematical Practices 8 Look for and express regularity in repeated reasoning. CCSS
You determined whether a sequence was geometric. Use geometric sequences. Find sums of geometric series. Then/Now
geometric means geometric series Vocabulary
Concept
an = a1 ● rn–1 Formula for the nth term Find the nth Term Find the sixth term of a geometric sequence for which a1 = –3 and r = –2. an = a1 ● rn–1 Formula for the nth term a6 = –3 ● (–2)6–1 n = 6, a1 = –3, r = –2 a6 = –3 ● (–32) (–2)5 = –32 a6 = 96 Multiply. Answer: Example 1
an = a1 ● rn–1 Formula for the nth term Find the nth Term Find the sixth term of a geometric sequence for which a1 = –3 and r = –2. an = a1 ● rn–1 Formula for the nth term a6 = –3 ● (–2)6–1 n = 6, a1 = –3, r = –2 a6 = –3 ● (–32) (–2)5 = –32 a6 = 96 Multiply. Answer: The sixth term is 96. Example 1
What is the fifth term of a geometric sequence for which a1 = 6 and r = 2? B. 96 C. 160 D. 384 Example 1
What is the fifth term of a geometric sequence for which a1 = 6 and r = 2? B. 96 C. 160 D. 384 Example 1
r = 10 ÷ 5 or 2; 5 is the first term. Write an Equation for the nth Term A. Write an equation for the nth term of the geometric sequence below. 5, 10, 20, 40, … r = 10 ÷ 5 or 2; 5 is the first term. an = a1 ● rn–1 nth term of a geometric sequence an = 5 ● 2n–1 a1 = 5 and r = 2 Answer: Example 2A
r = 10 ÷ 5 or 2; 5 is the first term. Write an Equation for the nth Term A. Write an equation for the nth term of the geometric sequence below. 5, 10, 20, 40, … r = 10 ÷ 5 or 2; 5 is the first term. an = a1 ● rn–1 nth term of a geometric sequence an = 5 ● 2n–1 a1 = 5 and r = 2 Answer: an = 5 ● 2n–1 Example 2A
an = a1 ● rn–1 nth term of a geometric sequence Write an Equation for the nth Term B. Write an equation for the nth term of the geometric sequence below. a5 = 4 and r = 3 Step 1 Find a1. an = a1 ● rn–1 nth term of a geometric sequence 4 = a1 ● 35–1 an = 4, n = 5, and r = 3 4 = a1 ● 81 Evaluate the power. Divide each side by 81. Example 2B
Step 2 Write the equation. Write an Equation for the nth Term Step 2 Write the equation. an = a1 ● rn–1 nth term of a geometric sequence Answer: Example 2B
Step 2 Write the equation. Write an Equation for the nth Term Step 2 Write the equation. an = a1 ● rn–1 nth term of a geometric sequence Answer: Example 2B
A. Write an equation for the nth term of the geometric sequence below A. an = 5 ● 3n–1 B. an = 3 ● 15n–1 C. an = 3 ● 5n–1 D. an = 5 ● 5n–1 Example 2A
A. Write an equation for the nth term of the geometric sequence below A. an = 5 ● 3n–1 B. an = 3 ● 15n–1 C. an = 3 ● 5n–1 D. an = 5 ● 5n–1 Example 2A
B. Write an equation for the nth term of the geometric sequence below B. Write an equation for the nth term of the geometric sequence below. a4 = 48 and r = 2 A. an = 6 ● 2n–1 B. an = 48 ● 2n–1 C. an = 12 ● 2n–1 D. an = 2 ● 6n–1 Example 2B
B. Write an equation for the nth term of the geometric sequence below B. Write an equation for the nth term of the geometric sequence below. a4 = 48 and r = 2 A. an = 6 ● 2n–1 B. an = 48 ● 2n–1 C. an = 12 ● 2n–1 D. an = 2 ● 6n–1 Example 2B
Find three geometric means between 3.12 and 49.92. Find Geometric Means Find three geometric means between 3.12 and 49.92. Step 1 Since there are three terms between the first and last term, there are 3 + 2 or 5 total terms, so n = 5. Step 2 Find r. an = a1rn–1 Formula for the nth term 49.92 = 3.12r5–1 n = 5, a1 = 3.12, a5 = 49.92 16 = r4 Divide each side by 3.12. ±2 = r Take the 4th root of each side. Example 3
Step 3 Use r to find the three geometric means. Find Geometric Means Step 3 Use r to find the three geometric means. There are two possible common ratios, so there are two possible sets of geometric means. Use each value of r to find three geometric means. r = 2 r = –2 a2 = 3.12(2) or 6.24 a2 = 3.12(–2) or –6.24 a3 = 6.24(2) or 12.48 a3 = –6.24(–2) or 12.48 a4 = 12.48(2) or 24.96 a4 = 12.48(–2) or –24.96 Answer: Example 3
Step 3 Use r to find the three geometric means. Find Geometric Means Step 3 Use r to find the three geometric means. There are two possible common ratios, so there are two possible sets of geometric means. Use each value of r to find three geometric means. r = 2 r = –2 a2 = 3.12(2) or 6.24 a2 = 3.12(–2) or –6.24 a3 = 6.24(2) or 12.48 a3 = –6.24(–2) or 12.48 a4 = 12.48(2) or 24.96 a4 = 12.48(–2) or –24.96 Answer: The geometric means are 6.24, 12.48, and 24.96, or –6.24, 12.48, and –24.96. Example 3
Find three geometric means between 12 and 0.75. A. 6, 3, 1.5 or –6, 3, –1.5 B. 6, 3, 1.5 or 6, –3, 1.5 C. 6, 3, 1.5 D. –6, 3, –1.5 Example 3
Find three geometric means between 12 and 0.75. A. 6, 3, 1.5 or –6, 3, –1.5 B. 6, 3, 1.5 or 6, –3, 1.5 C. 6, 3, 1.5 D. –6, 3, –1.5 Example 3
Concept
Find the Sum of a Geometric Series MUSIC Julian hears a song by a new band. He E-mails a link for the band’s Web site to five of his friends. They each forward the link to five of their friends. The link is forwarded again following the same pattern. How many people will receive the link on the sixth round of E-mails? Five E-mails are sent in the first round and there are 6 rounds of E-mails. So, a1 = 5, r = 5, and n = 6. Sum formula a1 = 5, r = 5, and n = 6. Example 4
Simplify the numerator and denominator. Find the Sum of a Geometric Series Simplify the numerator and denominator. Divide. Answer: Example 4
Simplify the numerator and denominator. Find the Sum of a Geometric Series Simplify the numerator and denominator. Divide. Answer: There will be 19,530 E-mails sent after 6 rounds. Example 4
ONLINE SHOPPING Ilani is the first one to be sent an online coupon for a 20% off sale on a retail Web site. She E-mails the coupon to three of her friends. They each forward the coupon to three of their friends. The link is forwarded again following the same pattern. How many people will receive the coupon on the seventh round of E-mails? A. 2,918 people B. 3,026 people C. 3,164 people D. 3,279 people Example 4
ONLINE SHOPPING Ilani is the first one to be sent an online coupon for a 20% off sale on a retail Web site. She E-mails the coupon to three of her friends. They each forward the coupon to three of their friends. The link is forwarded again following the same pattern. How many people will receive the coupon on the seventh round of E-mails? A. 2,918 people B. 3,026 people C. 3,164 people D. 3,279 people Example 4
Method 1 Since the sum is a geometric series, you can use the Sum in Sigma Notation Evaluate . Method 1 Since the sum is a geometric series, you can use the formula . Find a1, r, and n. In the first term, n = 1 and a1 = 3 ● 21 – 1 or 3. The base of the exponential function is r, so r = 2. There are 12 – 1 + 1 or 12 terms, so n = 12. Example 5
Sum formula n = 12, a1 = 3, r = 2 212 = 4096 S12 = 12,285 Simplify. Sum in Sigma Notation Sum formula n = 12, a1 = 3, r = 2 212 = 4096 S12 = 12,285 Simplify. Example 5
Sum in Sigma Notation Method 2 Find the terms by replacing n with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Then add. = 3(21–1) + 3(22–1) + 3(23–1) + 3(24–1) + 3(25–1) + 3(26–1) + 3(27–1) + 3(28–1) + 3(29–1) + 3(210–1) + 3(211–1) + 3(212–1) = 3(1) + 3(2) + 3(4) + 3(8) + 3(16) + 3(32) + 3(64) + 3(128) + 3(256) + 3(512) + 3(1024) + 3(2048) = 3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 + 768 + 1536 + 3072 + 6144 = 12,285 Example 5
Sum in Sigma Notation Answer: Example 5
Answer: The sum of the series is 12,285. Sum in Sigma Notation Answer: The sum of the series is 12,285. Example 5
Evaluate . A. 84 B. 200 C. 484 D. 600 Example 5
Evaluate . A. 84 B. 200 C. 484 D. 600 Example 5
Find a1 in a geometric series for which S8 = 765, n = 8, and r = 2. Find the First Term of a Series Find a1 in a geometric series for which S8 = 765, n = 8, and r = 2. Sum formula Sn = 765, r = 2, and n = 8 765 = Subtract. 765 = 255a1 Divide. 3 = a1 Divide each side by 255. Answer: Example 6
Find a1 in a geometric series for which S8 = 765, n = 8, and r = 2. Find the First Term of a Series Find a1 in a geometric series for which S8 = 765, n = 8, and r = 2. Sum formula Sn = 765, r = 2, and n = 8 765 = Subtract. 765 = 255a1 Divide. 3 = a1 Divide each side by 255. Answer: The first term of the series is 3. Example 6
Find a1 in a geometric series for which S6 = 364, n = 6, and r = 3. B. 1 C. 3 D. 4.5 Example 6
Find a1 in a geometric series for which S6 = 364, n = 6, and r = 3. B. 1 C. 3 D. 4.5 Example 6
End of the Lesson