Find the 13th term of the geometric sequence 7, 21, 63, …

Find the 13th term of the geometric sequence 7, 21, 63, ...

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  1. a=4

    r = \frac{-16}{4}r=4−16

    r = -4r=−4

    a_{13} = 4×(-4)^{(13-1)}a13=4×(−4)(13−1)

    a_{13} = 4×(-4)^{12}a13=4×(−4)12

    a_{13} = 4×16777216a13=4×16777216

    a_{13} = 67108864a13=67108864

  2. A geometric sequence is a sequence with a common ratio multiplied by previous term to give the next. Lets call x the common ratio, so we know that 4 was multiplied by x to give the second term, and so forth until it was multiplied again and again by x until the 16384 was reached which happens to be the 13th term, that means that 4x was multiplied by x 12 more times to give 16384, that is:
    4x^12 = 16384
    x^12 = 4096
    x = 4096^-12
    x = 4.48 x 10^-44
    that is the common ratio

  3. A

    Step-by-step explanation:

    You plug in 13 to n-1. to get -2^12. This equates to 4096, and you multiply that by 3

  4. [tex]\large \boxed{2}[/tex]

    Step-by-step explanation:

    The formula for the nth term of a geometric sequence is

    aₙ = a₁rⁿ⁻¹

    In your geometric sequence, a₁ = 4 and a₁₃ = 16 384.

    [tex]\begin{array}{rcl}16384 & = & 4r^{(13 - 1)}}\\16384 & = & 4r^{12}\\4096 & = & r^{12}\\3.6124 & = & 12 \log r\\0.30102 & = & \log r\\r & = & 10^{0.30102}\\ & = & \mathbf{2}\\\end{array}\\\text{The common ratio is $\large \boxed{\mathbf{2}}$}[/tex]

    Check:

    [tex]\begin{array}{rcl}16384 & = & 4(2)^{12}\\16384 & = & 4(4096)\\16384 & = & 16384\\\end{array}[/tex]

    It checks.

  5. 20, 480

    Step-by-step explanation:

    The sequence is just x multiplied by 2

    If you follow the sequence, on the 13th term, the number is 20,480.

  6. 4096

    Step-by-step explanation:

    The n th term of a geometric sequence is

    [tex]a_{n}[/tex] = a[tex](r)^{n-1}[/tex]

    where a is the first term and r the common ratio

    Here a = 1 and r = 2 ÷ 1 = 4 ÷ 2 = 2, thus

    [tex]a_{13}[/tex] = 1 × [tex]2^{12}[/tex] = 4096

  7. 1708984375

    Step-by-step explanation:

    In order to get from 7 to -35 you need to multiply by -5

    so that is what it is increasing by... a multiple of -5

    7, -35 , 175, -875, 4375, -21875, 109375, -546875, 2734375, -13671875, 68359375, -341796875, 1708984375

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