Mathematics

Question

Write the sum using summation notation, assuming the suggested pattern continues. 8 - 40 + 200 - 1000 + ...

2 Answer

  • Each successive term is -5 times the previous one, so the underlying sequence is

    [tex]a_n=-5a_{n-1}=(-5)^2a_{n-2}=(-5)^3a_{n-3}=\cdots=(-5)^{n-1}a_1=8(-5)^{n-1}[/tex]

    The sum can then be written as

    [tex]8-40+200-1000+\cdots=\displaystyle\sum_{n=1}^\infty8(-5)^{n-1}[/tex]
  • Answer:

    Step-by-step explanation:

    From the given information, the pattern is as:

    8 - 40 + 200 - 1000 + ...

    We can see that the successive term is  -5 times the previous one, therefore, using this, we can write it in the form of sequence:

    [tex]a_{n}=-5a_{n-1}=(-5)^2a_{n-2}=(-5)^3a_{n-3}=....=(-5)^{n-1}a_{1}=8(-5)^{n-1}[/tex].

    Therefore, the sum can be written as:

    8 - 40 + 200 - 1000 + ...=[tex]\sum_{n=1}^{\infty}8(-5)^{n-1}[/tex]

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