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Arithmetic and Geometric Sequence Solutions

Explore means and missing values in arithmetic and geometric series, with formulas, examples, and explanations for summation and sequences. Practice with practical exercises and learn about the Fibonacci sequence.

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Arithmetic and Geometric Sequence Solutions

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  1. Means for SeriesFind the value(s) missing from a list.

  2. Flashback to Arithmetic • 6, ___,14… • 6+14 =20 • 20/2=10 • 6,10,14 counting by 4 when you start at 6

  3. Arithmetic Series • Find the missing means. • 8,____, 18 2) 12, _____, 38 3) 20,___,____,____,36

  4. Check this out • Use the series: 5, 10, 20 • Take the first and third terms. • Multiply • Then square root. • = 10 the geometric mean between 5 and 20. • List the next two terms: 5, 10, 20, 40, 80 • Take 5 & 80, multiply then root to find the mean. • Then try 20 and 80.

  5. Formula for d.(last known value-first known value)/number of terms

  6. Find the missing terms of each geometric sequence • 1) 3, ____, 12 • 2) 45, ___, 180 • 3) 60, ___, (20/3) • 4) 9180, _____, 255 • 5) -20, ___, ___, ____, -1.25

  7. Let’s see something • Is 3+5 =5+3? • So 3+10+17+24+31+38= • 38+ 31+24+17+10+ 3? • first value + last value = 3+38=41 6*41 = 246 246/2= 123 Now we have the formula for the sum of an arithmetic sequence.

  8. Formula for a partial sum

  9. Try • Find the sum of the first 50 positive integers. • Find the sum: 7, 14, 21….98 • Find the sum: 10, 4, (-2), (-8)….(-50)

  10. Distributive Law • 10+5x= 5(_______________) • 5+5x = 5(____________) • -r =(____________)

  11. Clever Time • Take a geometric sequence Multiply by -r. Shift the list over one space and add. = 5 15 45 135 405 -r = -15 -45 -135 -405 -1215 -r = 5 -1215

  12. Sum of a Finite Geometric Series With r not equal to 1!

  13. Try • Geometric Series problems 1-5

  14. The sum of an infinite geometric series • If <1 = • If • Show a number line.

  15. Fibonacci Sequence

  16. A different pattern • Use two started values. • Start making the list using the formula

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