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Find the sum of the first 8 terms of the following series, to the nearest integer. 3,4,(16)/(3),dots Answer:

Find the sum of the first 88 terms of the following series, to the nearest integer.\newline3,4,163,… 3,4, \frac{16}{3}, \ldots \newlineAnswer:

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Full solution

Q. Find the sum of the first 88 terms of the following series, to the nearest integer.\newline3,4,163,… 3,4, \frac{16}{3}, \ldots \newlineAnswer:
  1. Determine Series Type: To find the sum of the first 88 terms of the series, we first need to determine if the series is arithmetic or geometric. We can do this by examining the relationship between consecutive terms.
  2. Identify Common Ratio: The second term is 44, which is 11 more than the first term, 33. However, the third term is 163\frac{16}{3}, which is not obtained by adding or subtracting the same value from the second term. Instead, it seems that each term is being multiplied by a common ratio to get the next term. To confirm this, let's find the ratio between the second and first term, and between the third and second term.
  3. Apply Sum Formula: The ratio between the second and first term is 43\frac{4}{3}. The ratio between the third and second term is (163)/4=43(\frac{16}{3}) / 4 = \frac{4}{3}. Since the ratio is consistent, this is a geometric series with a common ratio of 43\frac{4}{3}.
  4. Calculate (43)8(\frac{4}{3})^8: Now that we know it's a geometric series, we can use the formula for the sum of the first nn terms of a geometric series, which is Sn=a(1−rn)(1−r)S_n = \frac{a(1 - r^n)}{(1 - r)}, where aa is the first term, rr is the common ratio, and nn is the number of terms.
  5. Substitute Values: Let's plug in the values we know: a=3a = 3, r=43r = \frac{4}{3}, and n=8n = 8. So, S8=3(1−(43)8)/(1−43)S_8 = 3\left(1 - \left(\frac{4}{3}\right)^8\right) / \left(1 - \frac{4}{3}\right).
  6. Simplify Expression: First, we calculate (43)8(\frac{4}{3})^8. This is a calculation that typically requires a calculator, as it involves raising a fraction to a high power.
  7. Perform Subtraction: After calculating (43)8(\frac{4}{3})^8, we get approximately 655366561\frac{65536}{6561}. Now we can substitute this value back into the sum formula.
  8. Multiply by −9-9: Now we have S8=3(1−65536/6561)/(1−4/3)S_8 = 3(1 - 65536/6561) / (1 - 4/3). Since 1−4/31 - 4/3 is −1/3-1/3, we can simplify the denominator to −1/3-1/3.
  9. Round to Nearest Integer: Simplifying the expression, we get S8=3(1−655366561)×(−3)=−9(655366561−1)S_8 = 3(1 - \frac{65536}{6561}) \times (-3) = -9(\frac{65536}{6561} - 1).
  10. Round to Nearest Integer: Simplifying the expression, we get S8=3(1−655366561)×(−3)=−9(655366561−1)S_8 = 3(1 - \frac{65536}{6561}) \times (-3) = -9(\frac{65536}{6561} - 1).We need to subtract 11 from 655366561\frac{65536}{6561}, which gives us 655366561−65616561=65536−65616561\frac{65536}{6561} - \frac{6561}{6561} = \frac{65536 - 6561}{6561}.
  11. Round to Nearest Integer: Simplifying the expression, we get S8=3(1−655366561)×(−3)=−9(655366561−1)S_8 = 3(1 - \frac{65536}{6561}) \times (-3) = -9(\frac{65536}{6561} - 1).We need to subtract 11 from 655366561\frac{65536}{6561}, which gives us 655366561−65616561=65536−65616561\frac{65536}{6561} - \frac{6561}{6561} = \frac{65536 - 6561}{6561}.After performing the subtraction, we get 65536−65616561=589756561\frac{65536 - 6561}{6561} = \frac{58975}{6561}. Now we can multiply this by −9-9 to get the sum of the first 88 terms.
  12. Round to Nearest Integer: Simplifying the expression, we get S8=3(1−655366561)×(−3)=−9(655366561−1)S_8 = 3(1 - \frac{65536}{6561}) \times (-3) = -9(\frac{65536}{6561} - 1).We need to subtract 11 from 655366561\frac{65536}{6561}, which gives us 655366561−65616561=65536−65616561\frac{65536}{6561} - \frac{6561}{6561} = \frac{65536 - 6561}{6561}.After performing the subtraction, we get 65536−65616561=589756561\frac{65536 - 6561}{6561} = \frac{58975}{6561}. Now we can multiply this by −9-9 to get the sum of the first 88 terms.Multiplying −9-9 by 589756561\frac{58975}{6561} gives us −5307756561-\frac{530775}{6561}. This is the exact sum of the first 88 terms, but we need to round it to the nearest integer.
  13. Round to Nearest Integer: Simplifying the expression, we get S8=3(1−655366561)×(−3)=−9(655366561−1)S_8 = 3(1 - \frac{65536}{6561}) \times (-3) = -9(\frac{65536}{6561} - 1).We need to subtract 11 from 655366561\frac{65536}{6561}, which gives us 655366561−65616561=65536−65616561\frac{65536}{6561} - \frac{6561}{6561} = \frac{65536 - 6561}{6561}.After performing the subtraction, we get 65536−65616561=589756561\frac{65536 - 6561}{6561} = \frac{58975}{6561}. Now we can multiply this by −9-9 to get the sum of the first 88 terms.Multiplying −9-9 by 589756561\frac{58975}{6561} gives us −5307756561-\frac{530775}{6561}. This is the exact sum of the first 88 terms, but we need to round it to the nearest integer.Dividing 1111 by 1122 gives us approximately 1133. Rounding this to the nearest integer gives us 1144.

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