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Find the 10th term of the geometric sequence shown below.

x^(4),5x^(6),25x^(8),dots
Answer:

Find the 1010th term of the geometric sequence shown below.\newlinex4,5x6,25x8, x^{4}, 5 x^{6}, 25 x^{8}, \ldots \newlineAnswer:

Full solution

Q. Find the 1010th term of the geometric sequence shown below.\newlinex4,5x6,25x8, x^{4}, 5 x^{6}, 25 x^{8}, \ldots \newlineAnswer:
  1. Identify common ratio: Identify the common ratio (r) of the geometric sequence by dividing the second term by the first term.\newlineCalculation: r=5x6x4=5x64=5x2 r = \frac{5x^6}{x^4} = 5x^{6-4} = 5x^2
  2. Use nth term formula: Use the formula for the nth term of a geometric sequence, which is an=a1r(n1) a_n = a_1 \cdot r^{(n-1)} , where a1 a_1 is the first term and n n is the term number.\newlineCalculation: The first term a1 a_1 is x4 x^4 , the common ratio r r is 5x2 5x^2 , and the term number n n is 1010.
  3. Substitute values: Substitute the values into the formula to find the 1010th term.\newlineCalculation: a10=x4(5x2)101=x4(5x2)9 a_{10} = x^4 \cdot (5x^2)^{10-1} = x^4 \cdot (5x^2)^9
  4. Simplify expression: Simplify the expression by raising the common ratio to the 99th power.\newlineCalculation: (5x2)9=59(x2)9=59x18 (5x^2)^9 = 5^9 \cdot (x^2)^9 = 5^9 \cdot x^{18}
  5. Multiply terms: Multiply the first term x4 x^4 by the simplified common ratio 59x18 5^9 \cdot x^{18} .\newlineCalculation: a10=x459x18 a_{10} = x^4 \cdot 5^9 \cdot x^{18}
  6. Combine like terms: Combine the like terms by adding the exponents of x x .\newlineCalculation: a10=59x4+18=59x22 a_{10} = 5^9 \cdot x^{4+18} = 5^9 \cdot x^{22}
  7. Calculate coefficient: Calculate the value of 59 5^9 to get the coefficient of x22 x^{22} .\newlineCalculation: 59=1953125 5^9 = 1953125
  8. Write final expression: Write the final expression for the 1010th term.\newlineCalculation: a10=1953125x22 a_{10} = 1953125 \cdot x^{22}

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