Solving a Geometric Progression Problem with Given Differences
The problem involves determining the two possible values of the first term and the associated common ratio in a geometric progression, given the differences between terms. By following the steps outlined in the solution, the first term is calculated to be 1.2 or 3.2, and the common ratio is determined as 2. The calculations are illustrated step by step to aid understanding.
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A Geometric Progression Problem Source: Aidan Burrows (don t know where he got it from)
Note to Teacher Hand out the worksheets and if they struggle take them through the specific problem that follows. You can also set the related problem as an extension task (slide 9).
A Geometric Progression Problem The second term of a geometric progression is greater than the first term by 20. The fourth term is greater than the first by 35. Find the two possible values of the first term and associated common ratio.
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + 35 ??3= ? + 35 ?? = ? + 20 ? =? + 20 ?3=? + 35 ? ? ?+203 ?3 ?+35 ? = ? + 203= ?3+ 35?2 ?3+ 60?2+ 1200? + 8000 = ?3+ 35?2 25?2+ 1200? + 8000 = 0 ?2+ 48? + 320 = 0 ? + 40 ? + 8 = 0 ? = 40 , 8 ? =1 2, 3 2
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + 35 ??3= ? + 35 ?? = ? + 20 ? =? + 20 ?3=? + 35 ? ? ?+203 ?3 ?+35 ? = ? + 203= ?3+ 35?2 ?3+ 60?2+ 1200? + 8000 = ?3+ 35?2 25?2+ 1200? + 8000 = 0 ?2+ 48? + 320 = 0 ? + 40 ? + 8 = 0 ? = 40 , 8 ? =1 2, 3 2
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + 35 ??3= ? + 35 ?? = ? + 20 Alternatively: 20 35 (? 1) a = a = ?3 1 ? 1 20 ? 1= 35 ?3 1 20 ?3 1 = 35 ? 1 20 ?2+ ? + 1 ? 1 = 35 ? 1 20r2+ 20? + 20 = 35 4?2+ 4? 3 = 0 2r + 3 2? 1 = 0 ? =1 2, 3 2 ? = 40 , 8
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + 35 ??3= ? + 35 ?? = ? + 20 Alternatively: 20 35 (? 1) a = a = ?3 1 ? 1 20 ? 1= 35 ?3 1 20 ?3 1 = 35 ? 1 20 ?2+ ? + 1 ? 1 = 35 ? 1 20r2+ 20? + 20 = 35 4?2+ 4? 3 = 0 2r + 3 2? 1 = 0 ? =1 2 , 3 2 ? = 40 , 8
? = 40 ,8 Answer: ? =1 2 , 3 , respectively 2 Check: ? 1 2 3 4 ??2 ??3 ?? ? ?? 40 20 10 5 8 12 18 27
A Related Geometric Progression Problem The second term of a geometric progression is greater than the first term by 20. By what amount should the fourth term be greater than the first to yield a unique solution? Find the first term and common ratio.
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + ? ??3= ? + ? ?? = ? + 20 20 Q ? = ? = ?3 1 ? 1 20 ?3 1 = Q ? 1 20 ?2+ ? + 1 ? 1 = Q ? 1 20?2+ 20? + 20 = Q 20?2+ 20? + 20 ? = 0 ? = 20 400 4 20 20 ? 40 ? = 20 80? 1200 40 For a unique solution 80? 1200 = 0
? 1 2 3 4 ??2 ??3 ?? ? ?? = ? + 20 = ? + ? ??3= ? + ? ?? = ? + 20 20 Q ? = ? = ?3 1 ? 1 20 ?3 1 = Q ? 1 20 ?2+ ? + 1 ? 1 = Q ? 1 20?2+ 20? + 20 = Q 20?2+ 20? + 20 ? = 0 ? = 20 400 4 20 20 ? 40 ? = 20 80? 1200 40 For a unique solution 80? 1200 = 0
Q =1200 = 15 80 ? = 20 = 1 40 20 2 = 40 ? = 1 3 2 1 ? 1 2 3 4 ??2 10 ??3 ?? ? ?? 40 20 3 5 3 3 3 If we used ?in stead of 20 the general solution for ? would be: ?2 4? ? ? 2? 4?? 3?2 2? ? = ? = ? ? ?=3 4yields a single solution with ? = 2 3?and? = 1 Showing that 2 .
Q =1200 = 15 80 ? = 20 = 1 40 20 2 = 40 ? = 1 3 2 1 ? 1 2 3 4 ??2 10 ??3 ?? ? ?? 40 20 3 5 3 3 3 If we used ?in stead of 20 the general solution for ? would be: ?2 4? ? ? 2? 4?? 3?2 2? ? = ? = ? ? ?=3 4yields a single solution with ? = 2 3?and? = 1 Showing that 2 .
4?? 3?2 2? ? = ? 7? 4 3?2 ? 4? If ? ?=7 4then we get: ? = 2? ? = ? 7?2 3?2 2? ? = ? 2? 2? ? =1 2or? = 3 2 and ? = 2? or? = 2? 5, respectively. Do your answers to the original problem accord with the above? They should!
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