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Given the geometric sequence: 4,(24)/(17),(144)/(289)dots Find an explicit formula for a_(n). a_(n)= ◻ Find a_(7)= ◻

Given the geometric sequence: 4,2417,144289… 4, \frac{24}{17}, \frac{144}{289} \ldots \newlineFind an explicit formula for an a_{n} .\newlinean= a_{n}= \newline□ \square \newlineFind a7= a_{7}= □ \square

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Full solution

Q. Given the geometric sequence: 4,2417,144289… 4, \frac{24}{17}, \frac{144}{289} \ldots \newlineFind an explicit formula for an a_{n} .\newlinean= a_{n}= \newline□ \square \newlineFind a7= a_{7}= □ \square
  1. Identify common ratio: Identify the common ratio rr by dividing the second term by the first term.r=2417/4r = \frac{24}{17} / 4r=2417×14r = \frac{24}{17} \times \frac{1}{4}r=617r = \frac{6}{17}
  2. Write explicit formula: Write the explicit formula for the nth term of a geometric sequence: an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}.\newlineHere, a1=4a_1 = 4 and r=617r = \frac{6}{17}.\newlineSo, an=4⋅(617)(n−1)a_n = 4 \cdot \left(\frac{6}{17}\right)^{(n-1)}
  3. Find a7a_7: Find a7a_7 by substituting n=7n = 7 into the formula.\newlinea7=4×(617)7−1a_7 = 4 \times \left(\frac{6}{17}\right)^{7-1}\newlinea7=4×(617)6a_7 = 4 \times \left(\frac{6}{17}\right)^6
  4. Calculate a7a_7: Calculate a7a_7.
    a7=4×(617)6a_7 = 4 \times \left(\frac{6}{17}\right)^6
    a7=4×(4665624137569)a_7 = 4 \times \left(\frac{46656}{24137569}\right)
    a7=18662424137569a_7 = \frac{186624}{24137569}

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