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Find the sum of the first 9 terms of the following series, to the nearest integer. 2,(5)/(2),(25)/(8),dots Answer:

Find the sum of the first 99 terms of the following series, to the nearest integer.\newline2,52,258,… 2, \frac{5}{2}, \frac{25}{8}, \ldots \newlineAnswer:

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Q. Find the sum of the first 99 terms of the following series, to the nearest integer.\newline2,52,258,… 2, \frac{5}{2}, \frac{25}{8}, \ldots \newlineAnswer:
  1. Identify pattern: Identify the pattern of the series.\newlineThe given series is 2,52,258,…2, \frac{5}{2}, \frac{25}{8}, \ldots which looks like a geometric series. To confirm this, we need to find the common ratio by dividing the second term by the first term and the third term by the second term.\newlineCommon ratio (r)=522=54(r) = \frac{\frac{5}{2}}{2} = \frac{5}{4}\newlineAnd, r=25852=258×25=54r = \frac{\frac{25}{8}}{\frac{5}{2}} = \frac{25}{8} \times \frac{2}{5} = \frac{5}{4}\newlineSince the common ratio is the same for these terms, we can confirm that this is a geometric series with a common ratio of 54\frac{5}{4}.
  2. Use formula for sum: Use the formula for the sum of the first nn terms of a geometric series.\newlineThe sum of the first nn terms of a geometric series is given by the formula:\newlineSn=a⋅(1−rn)/(1−r)S_n = a \cdot (1 - r^n) / (1 - r), where aa is the first term, rr is the common ratio, and nn is the number of terms.\newlineIn this case, a=2a = 2, r=5/4r = 5/4, and n=9n = 9.
  3. Substitute and calculate: Substitute the values into the formula and calculate the sum.\newlineS9=2×(1−(54)9)/(1−54)S_9 = 2 \times (1 - (\frac{5}{4})^9) / (1 - \frac{5}{4})\newlineSince 54\frac{5}{4} is greater than 11, the series is divergent, and we need to adjust the formula to account for this:\newlineS9=2×((54)9−1)/(54−1)S_9 = 2 \times ((\frac{5}{4})^9 - 1) / (\frac{5}{4} - 1)\newlineS9=2×((54)9−1)/(14)S_9 = 2 \times ((\frac{5}{4})^9 - 1) / (\frac{1}{4})\newlineNow, calculate (54)9(\frac{5}{4})^9:\newline(54)9=5949(\frac{5}{4})^9 = \frac{5^9}{4^9}
  4. Calculate 595^9 and 494^9: Calculate 595^9 and 494^9.\newline59=19531255^9 = 1953125\newline49=2621444^9 = 262144\newlineNow we can continue with the calculation of S9S_9:\newlineS9=2×(1953125262144−1)/(14)S_9 = 2 \times (\frac{1953125}{262144} - 1) / (\frac{1}{4})
  5. Simplify expression: Simplify the expression.\newlineS9=2×(1953125262144−262144262144)/(14)S_9 = 2 \times (\frac{1953125}{262144} - \frac{262144}{262144}) / (\frac{1}{4})\newlineS9=2×(1953125−262144)/262144/(14)S_9 = 2 \times (1953125 - 262144) / 262144 / (\frac{1}{4})\newlineS9=2×(1690981262144)×4S_9 = 2 \times (\frac{1690981}{262144}) \times 4\newlineS9=2×169098165536S_9 = 2 \times \frac{1690981}{65536}\newlineS9=338196265536S_9 = \frac{3381962}{65536}
  6. Divide for sum: Divide to find the sum to the nearest integer.\newlineS9≈338196265536S_9 \approx \frac{3381962}{65536}\newlineS9≈51.6S_9 \approx 51.6\newlineTo the nearest integer, the sum is approximately 5252.

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