1. 1 ARITHMETIC AND GEOMETRIC SERIES Standards 22 and 23 Sum of Infinite Geometric Series: Problems Sum of Arithmetic Series: Problems Geometric Sequence: Problems Arithmetic Sequence: Problems Sum of Geometric Series: Problems END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2. 2 STANDARD 22: Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. STANDARD 23: Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3. 3 ESTÁNDAR 22: Los estudiantes encuentran términos generales y sumas de series aritméticas y de series geométricas finitas e infinitas. ESTÁNDAR 23: Los estudiantes derivan las fórmulas de suma para series aritméticas y series geométricas finitas e infinitas. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4. 4 Standards 22 and 23 Find the indicated term in each arithmetic sequence. 1 a for a = 6, and d=4. 16 a = a + (n - 1)d n 1 n= 16 16 a = 6 + (16-1)(4) = 6 + 15(4) = 6 + 60 = 66 a for 2, 6, 10, 14,… 12 a = a + (n - 1)d n 1 n= 12 12 a = 2 + (12-1)(4) = 2 + 11(4) = 2 + 44 = 46 d= 6-2 =4=4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5. 5 Standards 22 and 23 Find the missing terms in this arithmetic sequence. 6, __, __, __, -6 a = a + (n - 1)d n 1 1st 5th 1 a =6 5 a = -6 n= 5 -6 = 6 + (5-1)d -6 = 6 + 4d -6 -12 = 4d 4 d = -3 2 a = 6-3 =3=3 3 a = 3-3 =0=0 4 a = 0-3 = -3 3 0-3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6. 6 Standards 22 and 23 Find the missing terms in this arithmetic sequence. __, __, 12, __, 22 a = a + (n - 1)d n 1 3rd 5th 3 a =12 5 a = 22 n= 5 2 a = 2+5 =7=7 4 a = 12+5 = 17 d = 22-12 2 = 5 22 = a + (5-1)(5) 1 22 = a + (4)(5) 1 22 = a + 20 1 -20 1 a = 2 7 17 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7. 7 Standards 22 and 23 Find S for this arithmetic series described. n a = 14, a =210, n= 15. 1 n S = (a + a ) n 1 n n 2 S = (14 + 210) 15 2 S = (224) 15 2 S = (7.5)(224) 15 S = 1680 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8. 8 Standards 22 and 23 Find S for this arithmetic series described. n S = (a + a ) n 1 n n 2 with and a = a + (n - 1)d n 1 S = (a + a + (n-1)d) n 1 1 n 2 S = (2a + (n-1)d) n 1 n 2 a = 18, n= 20, d = 3 1 S = (2(18) + (20-1)(3)) 20 2 S = (36 + (19)(3)) 20 2 S = (36 + 57) 20 2 S = (93) 20 2 S = (10)(93) 20 S = 930 20 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9. 9 Standards 22 and 23 Find the first three terms of this arithmetic series. a = 5, a =61, S = 495. 1 n n S = (a + a ) n 1 n n 2 495 = (5 + 61) n 2 495 = (66) n 2 495 = 33n 33 n= 15 a = a + (n - 1)d n 1 61 = 5 + (15-1)d 61 = 5 + 14d -5 56 = 14d 14 d = 4 2 a = 5+4 =9=9 3 a = 9+4 = 13 First let’s find n: Now let’s find d: Finally let’s find second and third terms: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10. 10 Standards 22 and 23 Find the nth term of each geometric sequence. a = -6, r=3, n=6 1 a = a r 1n n-1 a = (-6)(3) 6 6-1 a = (-6)(3) 6 5 a = (-6)(243) 6 a = -1458 6 a = 4, r= -2, n=8 3 First let’s find a 1 4 -2 a = 2 = -2 -2 a = 1 = 1 Now using a = a r 1n n-1 a = (1)(-2) 8 8-1 a = (1)(-2) 8 7 a = (1)(-128) 8 a = -128 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11. 11 Standards 22 and 23 Find the nth term of this geometric sequence. First let’s find a 1 a = 2 Now using a = a r 1n n-1 a =, r=, n=9 3 4 5 2 3 4 5 2 3 = (4)(3) (2)(5) 6 5 = a = 1 6 5 2 3 = (6)(3) (2)(5) 9 5 = a = ( )( ) 9 9-1 9 5 2 3 a = ( )( ) 9 8 9 5 2 3 9 9 5 256 6561 a = 9 256 3645 a = 9 2304 32805.... 9 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12. 12 Standards 22 and 23 Find the sum of this geometric series. 7 + 21 + 63 + … to eight terms First find r: 21 7 r= = 3 a =7, r=3, n=8 1 S = a - a r 11 n 1 – r n S = 7 – (7)(3) 8 1 - 3 8 S = 7 – (7)(6561) -2 8 S = 7 – 45927 -2 8 S = -45920 -2 8 S = 22960 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13. 13 Standards 22 and 23 Find the sum of this geometric series. 3 + -6 + 12 + … to 7 terms First find r: -6 3 r= = -2 a =3, r=-2, n=7 1 S = a - a r 11 n 1 – r n S = 3 – (3)(-2) 7 1 - -2 7 S = 3 – (3)(-128) 3 7 S = 3 + 384 3 7 S = 387 3 7 S = 129 7 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14. 14 Standards 22 and 23 Find the sum of this infinite geometric series, if it exists. a =, r= 1 2 3 1 2 Since |r|<1 it has a sum S= a 1 1 - r S = 2 3 1 - 1 2 S = 2 3 1 2 = (2)(2) (3)(1) 4 3 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15. 15 Standards 22 and 23 Find the sum of this infinite geometric series, if it exists. -5 + 15 + -45 + 135 +… Lets find r first: 15 -5 r= = -3 Since |-3| > 1 this sum does not exist PRESENTATION CREATED BY SIMON PEREZ. All rights reserved