The 2nd,31st and the last terms of an AP are 5 and -3 respectively. Find the AP and its 30th term.

Answer

To solve the problem step by step, we will use the properties of an Arithmetic Progression (AP).

### Step 1: Understand the given information
We know:
- The second term of the AP (denoted as \( a + d \)) is 5.
- The 31st term of the AP (denoted as \( a + 30d \)) is -3.

### Step 2: Set up the equations
From the information given, we can write the following equations:
1. \( a + d = 5 \) (Equation 1)
2. \( a + 30d = -3 \) (Equation 2)

### Step 3: Solve for \( a \) in terms of \( d \)
From Equation 1, we can express \( a \) as:
\[ a = 5 - d \]

### Step 4: Substitute \( a \) in Equation 2
Now, substitute \( a \) in Equation 2:
\[ (5 - d) + 30d = -3 \]
This simplifies to:
\[ 5 - d + 30d = -3 \]
\[ 5 + 29d = -3 \]

### Step 5: Solve for \( d \)
Now, isolate \( d \):
\[ 29d = -3 - 5 \]
\[ 29d = -8 \]
\[ d = \frac{-8}{29} \]

### Step 6: Substitute \( d \) back to find \( a \)
Now substitute \( d \) back into the equation for \( a \):
\[ a = 5 - \left(\frac{-8}{29}\right) \]
\[ a = 5 + \frac{8}{29} \]
To combine these, convert 5 into a fraction:
\[ 5 = \frac{145}{29} \]
So,
\[ a = \frac{145}{29} + \frac{8}{29} = \frac{153}{29} \]

### Step 7: Write the AP
Now we have both \( a \) and \( d \):
- First term \( a = \frac{153}{29} \)
- Common difference \( d = \frac{-8}{29} \)

The AP can be expressed as:
\[ \text{AP} = a, a + d, a + 2d, \ldots \]
This gives:
\[ \text{AP} = \frac{153}{29}, \frac{145}{29}, \frac{137}{29}, \ldots \]

### Step 8: Find the 30th term
The 30th term of the AP is given by:
\[ a_{30} = a + 29d \]
Substituting the values:
\[ a_{30} = \frac{153}{29} + 29 \left(\frac{-8}{29}\right) \]
The \( 29 \) cancels out:
\[ a_{30} = \frac{153}{29} - 8 \]
Convert \( 8 \) into a fraction:
\[ 8 = \frac{232}{29} \]
Thus,
\[ a_{30} = \frac{153}{29} - \frac{232}{29} = \frac{153 - 232}{29} = \frac{-79}{29} \]

### Final Results
- The AP is: \( \frac{153}{29}, \frac{145}{29}, \frac{137}{29}, \ldots \)
- The 30th term is: \( \frac{-79}{29} \)