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Find the sum of the first 10 terms of the following series, to the nearest integer. 63,21,7,dots Answer:

Find the sum of the first 1010 terms of the following series, to the nearest integer.\newline63,21,7,… 63,21,7, \ldots \newlineAnswer:

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Full solution

Q. Find the sum of the first 1010 terms of the following series, to the nearest integer.\newline63,21,7,… 63,21,7, \ldots \newlineAnswer:
  1. Identify Sequence Type: Identify whether the given sequence is geometric or arithmetic.\newlineThe given sequence appears to reduce by a factor of 33 each time. This suggests that it is a geometric sequence.
  2. Find Common Ratio: Identify the common ratio of the geometric sequence.\newlineTo find the common ratio, divide the second term by the first term.\newline2163=13\frac{21}{63} = \frac{1}{3}\newlineCommon Ratio: 13\frac{1}{3}
  3. Calculate Sum Formula: Use the formula for the sum of the first nn terms of a geometric series to find the sum of the first 1010 terms.\newlineThe formula for the sum of a geometric series is Sn=a1×(1−rn)/(1−r)S_n = a_1 \times (1 - r^n) / (1 - r), where SnS_n is the sum of the first nn terms, a1a_1 is the first term, rr is the common ratio, and nn is the number of terms.
  4. Plug Values and Simplify: Plug the values into the formula to calculate the sum.\newlineS10=63×(1−(13)10)/(1−13)S_{10} = 63 \times (1 - (\frac{1}{3})^{10}) / (1 - \frac{1}{3})
  5. Calculate (1/3)10(1/3)^{10}: Simplify the expression to find the sum.\newlineS10=63×(1−(1/3)10)/(2/3)S_{10} = 63 \times (1 - (1/3)^{10}) / (2/3)\newlineTo simplify further, multiply the numerator and denominator by 33 to get rid of the fraction in the denominator.\newlineS10=63×3×(1−(1/3)10)/2S_{10} = 63 \times 3 \times (1 - (1/3)^{10}) / 2
  6. Perform Final Calculation: Calculate the value of (13)10(\frac{1}{3})^{10} to find the sum.\newline(13)10(\frac{1}{3})^{10} is a very small number, so for the purpose of finding the sum to the nearest integer, we can approximate it to 00.\newlineS10≈63×3×(1−0)/2S_{10} \approx 63 \times 3 \times (1 - 0) / 2
  7. Perform Final Calculation: Calculate the value of (13)10(\frac{1}{3})^{10} to find the sum.(13)10(\frac{1}{3})^{10} is a very small number, so for the purpose of finding the sum to the nearest integer, we can approximate it to 00.S10≈63×3×(1−0)/2S_{10} \approx 63 \times 3 \times (1 - 0) / 2Perform the final calculation.S10≈63×3/2S_{10} \approx 63 \times 3 / 2S10≈189/2S_{10} \approx 189 / 2S10≈94.5S_{10} \approx 94.5Since we need to find the sum to the nearest integer, we round 94.594.5 to 9595.

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