Get the answers you need from a community of experts on IDNLearn.com. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
To determine which formula matches the given sequence [tex]\(\frac{3}{2}, 1, \frac{11}{10}, \frac{18}{14}, \frac{27}{18}, \ldots\)[/tex], let's examine each option by calculating the first few terms using each provided formula.
given formulas:
(A) [tex]\( a_n = \frac{n^2 + 2}{4n + 2} \)[/tex]
(B) [tex]\( a_n = \frac{n^2 + 2}{4n - 2} \)[/tex]
(C) [tex]\( a_n = \frac{(n + 2)^2}{4n - 2} \)[/tex]
(D) [tex]\( a_n = \frac{n^2 - 2}{4n - 2} \)[/tex]
We will calculate the first five terms for each formula and compare them with the sequence provided: [tex]\(\frac{3}{2}, 1, \frac{11}{10}, \frac{18}{14}, \frac{27}{18}\)[/tex].
Option A:
[tex]\[ a_n = \frac{n^2 + 2}{4n + 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 + 2}{4 \cdot 1 + 2} = \frac{3}{6} = \frac{1}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 + 2}{4 \cdot 2 + 2} = \frac{6}{10} = \frac{3}{5} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 + 2}{4 \cdot 3 + 2} = \frac{11}{14} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 + 2}{4 \cdot 4 + 2} = \frac{18}{18} = 1 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 + 2}{4 \cdot 5 + 2} = \frac{27}{22} \][/tex]
These values do not match the given sequence. So, \textbf{Option A is not correct}.
Option B:
[tex]\[ a_n = \frac{n^2 + 2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 + 2}{4 \cdot 1 - 2} = \frac{3}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 + 2}{4 \cdot 2 - 2} = \frac{6}{6} = 1 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 + 2}{4 \cdot 3 - 2} = \frac{11}{10} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 + 2}{4 \cdot 4 - 2} = \frac{18}{14} = \frac{9}{7} \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 + 2}{4 \cdot 5 - 2} = \frac{27}{18} = \frac{3}{2} \][/tex]
These values match the given sequence exactly. So, \textbf{Option B is correct}.
Option C:
[tex]\[ a_n = \frac{(n + 2)^2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{(1 + 2)^2}{4 \cdot 1 - 2} = \frac{9}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{(2 + 2)^2}{4 \cdot 2 - 2} = \frac{16}{6} = \frac{8}{3} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{(3 + 2)^2}{4 \cdot 3 - 2} = \frac{25}{10} = \frac{5}{2} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{(4 + 2)^2}{4 \cdot 4 - 2} = \frac{36}{14} = \frac{18}{7} \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{(5 + 2)^2}{4 \cdot 5 - 2} = \frac{49}{18} \][/tex]
These values do not match the given sequence. So, \textbf{Option C is not correct}.
Option D:
[tex]\[ a_n = \frac{n^2 - 2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 - 2}{4 \cdot 1 - 2} = \frac{-1}{2} = -\frac{1}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 - 2}{4 \cdot 2 - 2} = \frac{2}{6} = \frac{1}{3} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 - 2}{4 \cdot 3 - 2} = \frac{7}{10} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 - 2}{4 \cdot 4 - 2} = \frac{14}{14} = 1 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 - 2}{4 \cdot 5 - 2} = \frac{23}{18} \][/tex]
These values do not match the given sequence. So, \textbf{Option D is not correct}.
Therefore, the correct formula for the [tex]\( n \)[/tex]th term in the sequence [tex]\(\frac{3}{2}, 1, \frac{11}{10}, \frac{18}{14}, \frac{27}{18}, \ldots\)[/tex] is given by option [tex]\( B \)[/tex]:
[tex]\[ \boxed{\frac{n^2 + 2}{4n - 2}} \][/tex]
given formulas:
(A) [tex]\( a_n = \frac{n^2 + 2}{4n + 2} \)[/tex]
(B) [tex]\( a_n = \frac{n^2 + 2}{4n - 2} \)[/tex]
(C) [tex]\( a_n = \frac{(n + 2)^2}{4n - 2} \)[/tex]
(D) [tex]\( a_n = \frac{n^2 - 2}{4n - 2} \)[/tex]
We will calculate the first five terms for each formula and compare them with the sequence provided: [tex]\(\frac{3}{2}, 1, \frac{11}{10}, \frac{18}{14}, \frac{27}{18}\)[/tex].
Option A:
[tex]\[ a_n = \frac{n^2 + 2}{4n + 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 + 2}{4 \cdot 1 + 2} = \frac{3}{6} = \frac{1}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 + 2}{4 \cdot 2 + 2} = \frac{6}{10} = \frac{3}{5} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 + 2}{4 \cdot 3 + 2} = \frac{11}{14} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 + 2}{4 \cdot 4 + 2} = \frac{18}{18} = 1 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 + 2}{4 \cdot 5 + 2} = \frac{27}{22} \][/tex]
These values do not match the given sequence. So, \textbf{Option A is not correct}.
Option B:
[tex]\[ a_n = \frac{n^2 + 2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 + 2}{4 \cdot 1 - 2} = \frac{3}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 + 2}{4 \cdot 2 - 2} = \frac{6}{6} = 1 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 + 2}{4 \cdot 3 - 2} = \frac{11}{10} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 + 2}{4 \cdot 4 - 2} = \frac{18}{14} = \frac{9}{7} \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 + 2}{4 \cdot 5 - 2} = \frac{27}{18} = \frac{3}{2} \][/tex]
These values match the given sequence exactly. So, \textbf{Option B is correct}.
Option C:
[tex]\[ a_n = \frac{(n + 2)^2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{(1 + 2)^2}{4 \cdot 1 - 2} = \frac{9}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{(2 + 2)^2}{4 \cdot 2 - 2} = \frac{16}{6} = \frac{8}{3} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{(3 + 2)^2}{4 \cdot 3 - 2} = \frac{25}{10} = \frac{5}{2} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{(4 + 2)^2}{4 \cdot 4 - 2} = \frac{36}{14} = \frac{18}{7} \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{(5 + 2)^2}{4 \cdot 5 - 2} = \frac{49}{18} \][/tex]
These values do not match the given sequence. So, \textbf{Option C is not correct}.
Option D:
[tex]\[ a_n = \frac{n^2 - 2}{4n - 2} \][/tex]
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = \frac{1^2 - 2}{4 \cdot 1 - 2} = \frac{-1}{2} = -\frac{1}{2} \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = \frac{2^2 - 2}{4 \cdot 2 - 2} = \frac{2}{6} = \frac{1}{3} \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = \frac{3^2 - 2}{4 \cdot 3 - 2} = \frac{7}{10} \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = \frac{4^2 - 2}{4 \cdot 4 - 2} = \frac{14}{14} = 1 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = \frac{5^2 - 2}{4 \cdot 5 - 2} = \frac{23}{18} \][/tex]
These values do not match the given sequence. So, \textbf{Option D is not correct}.
Therefore, the correct formula for the [tex]\( n \)[/tex]th term in the sequence [tex]\(\frac{3}{2}, 1, \frac{11}{10}, \frac{18}{14}, \frac{27}{18}, \ldots\)[/tex] is given by option [tex]\( B \)[/tex]:
[tex]\[ \boxed{\frac{n^2 + 2}{4n - 2}} \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Trust IDNLearn.com for all your queries. We appreciate your visit and hope to assist you again soon.
