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11.3: Geometric Sequences & Series. n th Term of an Geometric Sequence: a n = a 1 r (n – 1)

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Presentation on theme: "11.3: Geometric Sequences & Series. n th Term of an Geometric Sequence: a n = a 1 r (n – 1)"— Presentation transcript:

1 11.3: Geometric Sequences & Series

2 n th Term of an Geometric Sequence: a n = a 1 r (n – 1)

3 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2.

4 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1)

5 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1)

6 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)r (8 – 1)

7 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2) (8 – 1)

8 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2 7 ) a 8 = (-3)(-2 7 )

9 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2 7 ) a 8 = (-3)(-2 7 ) a 8 = (-3)(-128) a 8 = (-3)(-128)

10 n th Term of an Geometric Sequence: a n = a 1 r (n – 1) Ex. 2 Determine the following: a) Find the eighth term of a geometric sequence for which a 1 = -3 and r = -2. a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 8 = a 1 r (8 – 1) a 8 = a 1 r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)r (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2) (8 – 1) a 8 = (-3)(-2 7 ) a 8 = (-3)(-2 7 ) a 8 = (-3)(-128) a 8 = (-3)(-128) a 8 = 384 a 8 = 384

11 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, …

12

13 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … ·4 ·4

14 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4

15 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1)

16 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)r (n – 1) a n = (3)r (n – 1)

17 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)(4) (n – 1) a n = (3)(4) (n – 1)

18 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1

19 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = 3.

20 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1

21 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a n = a 1 r (n – 1)

22 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1)

23 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1)

24 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1)

25 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a 1 3 3

26 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1

27 b) Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, … r = 4 r = 4 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a n = (3)4 n – 1 a n = (3)4 n – 1 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1

28 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10

29 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1)

30 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1)

31 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)r (10 – 1)

32 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3) (10 – 1)

33 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3 9 ) a 10 = (4)(3 9 )

34 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3 9 ) a 10 = (4)(3 9 ) a 10 = (4)(19,683) a 10 = (4)(19,683)

35 c)Find the tenth term of a geometric sequence for which a 4 = 108 and r = Find a 1 : a 4 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 r (4 – 1) 108 = a 1 3 (4 – 1) 108 = a 1 3 (4 – 1) 108 = a = a = 27a = 27a 1 4 = a 1 4 = a 1 2. Now use a 1 = 4 and r = 3 to find a 10 a n = a 1 r (n – 1) a n = a 1 r (n – 1) a 10 = a 1 r (10 – 1) a 10 = a 1 r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)r (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3) (10 – 1) a 10 = (4)(3 9 ) a 10 = (4)(3 9 ) a 10 = (4)(19,683) a 10 = (4)(19,683) a 10 = 78,732 a 10 = 78,732

36 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r

37 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2.

38 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r

39 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r

40 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – r 15 ) S 15 = 2(1 – r 15 ) 1 – r 1 – r

41 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – r 15 ) S 15 = 2(1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – 2 15 ) S 15 = 2(1 – 2 15 ) 1 – 2 1 – 2

42 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – r 15 ) S 15 = 2(1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – 2 15 ) S 15 = 2(1 – 2 15 ) 1 – 2 1 – 2 S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) 1 – – 2 -1

43 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – r 15 ) S 15 = 2(1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – 2 15 ) S 15 = 2(1 – 2 15 ) 1 – 2 1 – 2 S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 1 – –

44 Sum of a Geometric Series The sum S n of the first n terms of a geometric series is given by the following: S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r Ex. 3 Find the sum of the first 15 terms of the geometric sequence in which a 1 = 2 and r = 2. S n = a 1 (1 – r n ) S n = a 1 (1 – r n ) 1 – r 1 – r S 15 = a 1 (1 – r 15 ) S 15 = a 1 (1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – r 15 ) S 15 = 2(1 – r 15 ) 1 – r 1 – r S 15 = 2(1 – 2 15 ) S 15 = 2(1 – 2 15 ) 1 – 2 1 – 2 S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 = 65,534 S 15 = 2(1 – 2 15 ) = 2(1 – 32,768) = -65,534 = 65,534 1 – –


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