arithmetic means.pptx, common ratio and sequence

arithmetic means.pptx, common ratio and sequence

1. Arithmetic Means Term insert in between the two given terms

2. Example 1: Insert 2 arithmetic means between 4 and 19. 4, ___, ___, 19 a n = a 1 + (n – 1) d 19 = 4 + (4 – 1)d 19 = 4 + (3)d 19 = 4 + 3d 19 - 4 = 3d 15 = 3d =   a 2 = 4 + (2-1)5 a 2 = 4 + (1)5 a 2 = 4 + 5 a 2 = 9 a 3 = 4 + (3-1)5 a 3 = 4 + (2)5 a 3 = 4 + 10 a 3 = 14 9 14 d = 5

3. Or the faster way

4. Example A: Insert 2 arithmetic means between 4 and 19. 4, ___, ___, 19 a 2 = 4 + 5 a 2 = 9 a 3 = 9 + 5 a 3 = 14 9 14 d =   d =   d =   d = 5 Or add 5 every after each term to find the next term

5. Example 2: Insert 3 arithmetic means between 54 and 6. 54, ___, ___, ___, 6 a n = a 1 + (n – 1) d 6 = 54 + (5 – 1)d 6 = 54 + (4)d 6 = 54 + 4d 6 - 54 = 4d -48 = 4d =   42 30 18 Add -12 every after each term to find the arithmetic means. d = -12

6. 42 30 18 Example B: Insert 3 arithmetic means between 54 and 6. 54, __, ___, ___, 6 d =   d =   d =   d = -12 Or add -12 every after each term to find the next term

7. Example 3: Insert 3 arithmetic means between 54 and 6. -4, __, ___, ___, 8 a n = a 1 + (n – 1) d 8 = -4 + (5 – 1)d 8 = -4 + (4)d 8 = -4 + 4d 8 + 4 = 4d 12 = 4d =   -1 2 5 Add 3 every after each term to find the arithmetic means. d = 3

8. Example 4: Insert 2 arithmetic means between 19 and -71. 19 , ___, ___, -71 a n = a 1 + (n – 1) d -71 = 19 + (4 – 1)d -71 = 19 + (3)d -71 = 19 + 3d -71 – 19 = 3d -90= 3d = d = -30   -11 -41 Add -30 every after each term to find the arithmetic means.

9. Example 5: Insert 3 arithmetic means between 54 and 6. , __, ___, ___,   a n = a 1 + (n – 1) d = + (5 – 1)d = + (4)d = + 4d       Add every after each term and simplify to find the arithmetic means.   - = 4d = 4d =   x = d = d d =    

10. Let’s do the shorter way to get the common difference then insert some arithmetic means

11. 6 30 18 Example C: Insert 6 arithmetic means between 54 and 6. 42, __, ___, ___, ___, ___, ___ -42 d =   d =   d =   d = -12 -6 -18 -30 Or add -12 every after each term to find the next term

12. Any clarifications?

13. Let’s test ourselves

14. Drill: 1) - , ___ , -   2) 10, ___, ___, ___ , 130 3) 19, ___, ___, −71, 4) −7.7, ____ , ____ , −16.4 -   40 70 100 −11 −41 −10.6 −13.5 d = -   d = 30 d = -30 d = -2.9

15. Example 2: Insert 3 arithmetic means between 54 and 6. 54, ___, ___, ___, 6 a n = a 1 + (n – 1) d 6 = 54 + (5 – 1)d 6 = 54 + (4)d 6 = 54 + 4d 6 - 54 = 4d -48 = 4d =   42 30 18 Add -12 every after each term to find the arithmetic means. d = -12