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 September 4th, 2017, 03:28 AM #1 Newbie   Joined: Sep 2017 From: Manila Posts: 1 Thanks: 0 Sequences and Series Hey! I need help with this question.. An arithmetic sequence has the first term -4 and common difference 1. A geometric sequence has the first term 8 and common difference 0.5. After how many terms would the sum of the arithmetic sequence exceed the sum of the geometric sequence? Thanks!
 September 4th, 2017, 04:18 AM #2 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 879 Thanks: 60 Math Focus: सामान्य गणित Like Arithmetic sequence has common difference, Geometric sequence has common ratio. A.P. = 4, 5, 6, 7, ........... G.P. = 8, 4, 2, 1, ........... Sum upto 3rd term, A.P. = 4 + 5 + 6 = 15 G.P. = 8 + 4 + 2 = 14 So, the sum of A.P. exceeds the sum of G.P. after third term.
 September 4th, 2017, 05:42 AM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,900 Thanks: 1094 Math Focus: Elementary mathematics and beyond It's stated that the A.P. has first term -4, not 4.
 September 7th, 2017, 09:09 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 So for the a.p., $\displaystyle a_n= -4+ n$, (the first term is $\displaystyle a_0$) and for the g.p. $\displaystyle b_n= 8(1/2)^n$ (again the first term is $\displaystyle b_0$. You want $\displaystyle -4+ n> 8(1/2)^n$. Clearly, the right side is positive so n must be at least 5. $\displaystyle 8(1/2)^3< 1$ so taking $n= 5$, $\displaystyle a_n= a_5= 1$ and $\displaystyle 8(1/2)^n= 8/8^5= 1/8^4= 0.000244140625$ so $\displaystyle a_5> b_5$. Last edited by skipjack; September 7th, 2017 at 11:33 AM.
September 7th, 2017, 11:01 AM   #5
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Quote:
 Originally Posted by kirtanadevaraj An arithmetic sequence has the first term -4 and common difference 1. A geometric sequence has the first term 8 and common difference 0.5. After how many terms would the sum of the arithmetic sequence exceed the sum of the geometric sequence?
arithmetic: a = 1st term, d = common difference, n = number of terms
geometric: b = 1st term, r = common ratio, n = number of terms

Formula for sum of 1st n terms:
arithmetic: (2a + d(n - 1)) / 2 [1]
geometric: b(r^n - 1) / (r - 1) [2]

[1] = [2] : solve for n

Try at Wolfram: you'll be told to get lost!!

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