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Arithmetic series worksheet

Try a few problems where you'll be asked to evaluate finite arithmetic series.

Practice evaluating arithmetic series

Problem 1

sum, start subscript, k, equals, 1, end subscript, start superscript, 275, end superscript, left parenthesis, minus, 5, k, plus, 12, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 2

The arithmetic sequence a, start subscript, i, end subscript is defined by the formula:
a, start subscript, 1, end subscript, equals, 2
a, start subscript, i, end subscript, equals, a, start subscript, i, minus, 1, end subscript, minus, 3
Find the sum of the first 335 terms in the sequence.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 3

Find the sum of minus, 50, plus, left parenthesis, minus, 44, right parenthesis, plus, left parenthesis, minus, 38, right parenthesis, plus, point, point, point, plus, 2038, plus, 2044.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to join the conversation?

  • aqualine seed style avatar for user Valerie Vargas
    Why do you always add one when solving for n?
    (17 votes)
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    • female robot grace style avatar for user beachhermit
      You have to add one because you're working out how many items there are in the series by counting how many hops it takes to get from the first to the last.

      Imagine there is a line of 5 squares on the ground. You are standing in the first. You hop from the first to the second: 1 hop. Then from the second to the third: 2 hops. Third to fourth: 3 hops. Fourth to fifth: 4 hops. You have to hop 4 times to get from your initial position in the first square to your final position in the fifth square. More generally, for each hop you take, the number of hops is always one less than the number of squares you've been standing in.

      This is what we do when we divide by the difference. We start out on the first item in the sequence, then work out how many times we have to add our constant to get to the final item. That show us how many times we've added.

      But, as (I hope) is shown in the example with the squares, there's always going to be one more in the series than the number of hops, because it doesn't count your starting position, only the movement between positions
      (44 votes)
  • leaf green style avatar for user Aaron Loertscher
    In problem #2 what does ai-1 refer too?
    (6 votes)
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    • blobby green style avatar for user InnocentRealist
      The "i-1" is a subscript. "i" is the "index", which tells you which term you are in. Another way to write it is "a(i-1)". "a(i-1) is the term before "a(i)". "a(i+2)" would be the 2nd term after "a(i)".

      When i is 1, what's i-1? when i is 335, what's i-1? What's i for the first term? The 335th term?

      0, 334, 1, 335.
      (8 votes)
  • orange juice squid orange style avatar for user Ian B
    In the first question, how do you get n=275?
    (1 vote)
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  • leaf green style avatar for user Muhammadaminbek
    Hi guys I need help! since long break I have forgotten math a little bit!
    The sum of five numbers in arithmetic progression is 40 and the sum of their squares is
    410; find the five numbers
    (2 votes)
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    • leaf green style avatar for user bhvima
      The five number are:
      a, (a + k), (a + 2k), (a + 3k), (a + 4k)

      a + (a + k) + (a + 2k) + (a + 3k) + (a + 4k) = 40
      5a + 10k = 40

      a^2 + (a + k)^2 + (a + 2k)^2 + (a + 3k)^2 + (a + 4k)^2 = 410
      5a^2 + 30k^2 + 20ak = 410

      We know that 5a + 10k = 40, then a = 8 - 2k

      5(8-2k)^2+30k^2+20(8-2k)k=410

      k = -3 or 3
      a = 14 or 2


      The numbers are either 14, 11, 8, 5, 2 or 2, 5, 8, 11, 14.

      Hope this helps.
      (4 votes)
  • blobby green style avatar for user Manash Kalita
    In problem after multipying -499 by 335 Im'm getting -15215. I guess I'm doing something wrong here.
    (2 votes)
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  • leafers ultimate style avatar for user Wesley
    In Problem #2. Can anyone explain the reasoning behind subtracting 1000 (EDIT: 1002, NOT 1000) from 2? Why would you want to "set it to zero"?
    (1 vote)
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    • aqualine tree style avatar for user Judith Gibson
      To find the sum of terms in an arithmetic series we need to :
      1. Find the first term (in this case, 2)
      2. Find the last term (in this case, - 1000)
      3. Take the average of their sum : (in this case, { 2 + - 1000) / 2 }
      That's why it looks like 1000 is being subtracted from 2.
      As for your "set it to zero" question, I can't see where that is in the video.
      (6 votes)
  • spunky sam blue style avatar for user Adarsh Rawat
    How to prove a^2(b+c) + b^2(a+c) + c^2(a+b) = (a+b+c)^3 * 2/9
    (2 votes)
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  • blobby green style avatar for user abibitton
    In problem #1:
    How come when I use this Formula: An= A1 + (n-1)d to find n it wont give me the right answer?
    It comes out to: (((-1,363)-7)/5) +1 = 273

    -1,363 = 7+(n-1)5 -> -1,370=(n-1)5 -> -274= n-1 -> here i should add one, but according the the formula I have to subtract, ending with 273.

    Am I missing a rule here? when its negative you add?
    (2 votes)
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    • blobby green style avatar for user InnocentRealist
      In this problem, what's the first term of the sequence?

      Right, it's 12-5 = 7. So the first term already has 5 subtracted from it. The subtraction starts with the first term, not the second. So, the formula is:

      "a(k) = 12 - 5k",
      not "a(k) = 12 - 5(k-1)".,

      because the sum is "sigma (k=1 to 275) of (-5k + 12)".
      (2 votes)
  • blobby green style avatar for user Bradley424
    how come in question 2, you're decreasing by -3 for each new term, but yet when you multiplied how many terms are in the sequence with the -3, you took out the negative in the -3 to make it 334 multiplied by 3 and have the answer come out to be 1002
    (1 vote)
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    • primosaur tree style avatar for user Bradley Reynolds
      What they did there is actually still correct. This is because they are not "decreasing by -3" but are actually just decreasing by 3, or alternatively increasing by -3. So you are right, 334 times -3 is -1002, which is why they do 2 - 1002. You will note they say "decreases by a total of 334 * 3".
      This way of writing it might be a little confusing but if you look at it carefully you can usually figure it out.
      (3 votes)
  • blobby green style avatar for user evan.gruber
    How do you get to the 350 part of the equation?
    (1 vote)
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