The nth term of a sequence in defined as follows. Find the first four terms :
(i) `a_(n)=3n+1 " " (ii) a_(n)=n^(2)+3 " " (iii) a_(n)=n(n+1) " " (iv) a_(n)=n+(1)/(n) " " (v) a_(n)=3^(n)`

Answer

To find the first four terms of each sequence defined by the given nth term formulas, we will evaluate each formula for n = 1, 2, 3, and 4.

### (i) \( a_n = 3n + 1 \)

1. **First Term (\( a_1 \))**:
\[ a_1 = 3(1) + 1 = 3 + 1 = 4 \]

2. **Second Term (\( a_2 \))**:
\[ a_2 = 3(2) + 1 = 6 + 1 = 7 \]

3. **Third Term (\( a_3 \))**:
\[ a_3 = 3(3) + 1 = 9 + 1 = 10 \]

4. **Fourth Term (\( a_4 \))**:
\[ a_4 = 3(4) + 1 = 12 + 1 = 13 \]

**First four terms**: 4, 7, 10, 13

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### (ii) \( a_n = n^2 + 3 \)

1. **First Term (\( a_1 \))**:
\[ a_1 = 1^2 + 3 = 1 + 3 = 4 \]

2. **Second Term (\( a_2 \))**:
\[ a_2 = 2^2 + 3 = 4 + 3 = 7 \]

3. **Third Term (\( a_3 \))**:
\[ a_3 = 3^2 + 3 = 9 + 3 = 12 \]

4. **Fourth Term (\( a_4 \))**:
\[ a_4 = 4^2 + 3 = 16 + 3 = 19 \]

**First four terms**: 4, 7, 12, 19

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### (iii) \( a_n = n(n + 1) \)

1. **First Term (\( a_1 \))**:
\[ a_1 = 1(1 + 1) = 1 \cdot 2 = 2 \]

2. **Second Term (\( a_2 \))**:
\[ a_2 = 2(2 + 1) = 2 \cdot 3 = 6 \]

3. **Third Term (\( a_3 \))**:
\[ a_3 = 3(3 + 1) = 3 \cdot 4 = 12 \]

4. **Fourth Term (\( a_4 \))**:
\[ a_4 = 4(4 + 1) = 4 \cdot 5 = 20 \]

**First four terms**: 2, 6, 12, 20

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### (iv) \( a_n = n + \frac{1}{n} \)

1. **First Term (\( a_1 \))**:
\[ a_1 = 1 + \frac{1}{1} = 1 + 1 = 2 \]

2. **Second Term (\( a_2 \))**:
\[ a_2 = 2 + \frac{1}{2} = 2 + 0.5 = \frac{5}{2} \]

3. **Third Term (\( a_3 \))**:
\[ a_3 = 3 + \frac{1}{3} = 3 + 0.3333 = \frac{10}{3} \]

4. **Fourth Term (\( a_4 \))**:
\[ a_4 = 4 + \frac{1}{4} = 4 + 0.25 = \frac{17}{4} \]

**First four terms**: 2, \( \frac{5}{2} \), \( \frac{10}{3} \), \( \frac{17}{4} \)

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### (v) \( a_n = 3^n \)

1. **First Term (\( a_1 \))**:
\[ a_1 = 3^1 = 3 \]

2. **Second Term (\( a_2 \))**:
\[ a_2 = 3^2 = 9 \]

3. **Third Term (\( a_3 \))**:
\[ a_3 = 3^3 = 27 \]

4. **Fourth Term (\( a_4 \))**:
\[ a_4 = 3^4 = 81 \]

**First four terms**: 3, 9, 27, 81

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