Determine the sum of the first 100 numbers. (arithmetic sequence)?

In an arithmetic sequence, the first three numbers are 100, 96 and 92.

Determine the sum of the first 100 numbers.

I only know that the answer is -9800, can someone show how I can get this answer?

7 Answers

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  • Mike G
    Lv 7
    6 months ago
    Favorite Answer

    a = 100

    d = -4

    100th term = 100-4*99 = -396

    Average term = (100-396)/2 = -98

    Sum = -98*100 = -9800

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  • Como
    Lv 7
    6 months ago

    Sn = (n/2) [ 2a + (n - 1) d ]

    S100 = 50 [ 2 + 99 x 1 ]

    S 100 = 50 x 101

    S 100 = 5050

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  • sepia
    Lv 7
    6 months ago

    100 + 96 + 92 + ...... + (100 - 4*99)

    = 100/2(100 + 100 - 4*99)

    = 50(200 - 396)

    = 100(-196)/2

    = -19600/2

    = -9800

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  • 6 months ago

    S(n) = n/2(2a + (n-1)d) is the general equation

    where

    n = 100

    a = 100

    d = -4

    Substituting

    S(100) = 100/2(2(100) + ( 100-1)(-4))

    S(100) = 50(200 + 99(-4)

    S(100) = 50(200 - 396)

    S(100) = 50(-196)

    S(100) = -9800 As required.

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  • 6 months ago

    Sn = (n/2)[2a + (n - 1)d]....where a is the first term, d is the common difference and n is the number of terms

    i.e. S₁₀₀ = (100/2)[200 - 4(99)]

    => S₁₀₀ = 50(-196)

    so, -9800

    :)>

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  • TomV
    Lv 7
    6 months ago

    A(n) = 100 - 4(n-1)

    A(1) = 100

    A(100) = 100 - 396 = -296

    Σ = number of terms multiplied by the average value of each term:

    = 100(A(1) + A(100))/2

    = 100(100-296)/2

    = 50(-196)

    Ans: Σ = -9800

    Or:

    Σ(104-4n) = 100(104) - 4Σn

    = 10400 - 400(101)/2

    = 10400 - 20200

    = -9800

    Either way the answer is the same, -9800

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  • S[n] = (n/2) * (t[1] + t[n])

    This holds true for all arithmetic sequences

    S[100] = (100/2) * (t[1] + t[100])

    S[100] = 50 * (t[1] + t[100])

    The first term is 100, or 100 - 0, or 104 - 4, or 104 - 4 * 1

    The 2nd term is 96, or 100 - 4, or 104 - 8, or 104 - 4 * 2

    The 3rd term is 92, or 104 - 12, or 104 - 4 * 3

    See a pattern?

    t[1] = 100

    t[100] = 104 - 4 * 100 = 104 - 400 = -296

    50 * (100 + (-296)) =>

    50 * (-196) =>

    50 * 2 * (-98) =>

    -98 * 100 =>

    -9800

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