If `nth` term of the series `25 + 29 +33+37......` and `3+4+6+9+13 + ........` are equal, then `n` equal

Options

A. 11
B. 12
C. 13
D. 14

Answer

To solve the problem, we need to find the value of \( n \) such that the \( n \)-th term of the first series is equal to the \( n \)-th term of the second series.

### Step 1: Identify the first series
The first series is:
\[ 25, 29, 33, 37, \ldots \]
This is an arithmetic series where the first term \( a = 25 \) and the common difference \( d = 4 \).

The \( n \)-th term \( T_n \) of an arithmetic series can be calculated using the formula:
\[ T_n = a + (n-1)d \]
Substituting the values:
\[ T_n = 25 + (n-1) \cdot 4 \]
\[ T_n = 25 + 4n - 4 \]
\[ T_n = 4n + 21 \]

### Step 2: Identify the second series
The second series is:
\[ 3, 4, 6, 9, 13, \ldots \]
To find the pattern, we can look at the differences:
- \( 4 - 3 = 1 \)
- \( 6 - 4 = 2 \)
- \( 9 - 6 = 3 \)
- \( 13 - 9 = 4 \)

The differences are \( 1, 2, 3, 4, \ldots \), which form an arithmetic series with a common difference of 1.

To find the \( n \)-th term of this series, we can express it as:
- The first term is \( 3 \).
- The second term is \( 3 + 1 = 4 \).
- The third term is \( 4 + 2 = 6 \).
- The fourth term is \( 6 + 3 = 9 \).
- The fifth term is \( 9 + 4 = 13 \).

The \( n \)-th term \( S_n \) can be expressed as:
\[ S_n = 3 + (1 + 2 + 3 + \ldots + (n-1)) \]
The sum of the first \( (n-1) \) natural numbers is given by:
\[ \frac{(n-1)n}{2} \]
Thus,
\[ S_n = 3 + \frac{(n-1)n}{2} \]

### Step 3: Set the \( n \)-th terms equal
Now, we set the two \( n \)-th terms equal to each other:
\[ 4n + 21 = 3 + \frac{(n-1)n}{2} \]

### Step 4: Solve for \( n \)
First, simplify the equation:
\[ 4n + 21 = 3 + \frac{n^2 - n}{2} \]
Multiply through by 2 to eliminate the fraction:
\[ 2(4n + 21) = 2(3) + (n^2 - n) \]
\[ 8n + 42 = 6 + n^2 - n \]
Rearranging gives:
\[ n^2 - 9n - 36 = 0 \]

### Step 5: Factor the quadratic equation
Now we can factor the quadratic:
\[ (n - 12)(n + 3) = 0 \]
Thus, the solutions are:
\[ n = 12 \quad \text{or} \quad n = -3 \]

Since \( n \) must be a positive integer (as it represents a term number), we take:
\[ n = 12 \]

### Final Answer
The value of \( n \) is:
\[ n = 12 \]

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