Answer :
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]
