`99^(th)` term of the series `2+7+14+23...`

Options

A. 9998
B. 9999
C. 10000
D. 100000

Answer

To find the 99th term of the series \(2 + 7 + 14 + 23 + \ldots\), we first need to identify the pattern in the series.

### Step 1: Identify the pattern in the series
The given series is:
- \(2\)
- \(7\)
- \(14\)
- \(23\)

Now, let's find the differences between consecutive terms:
- \(7 - 2 = 5\)
- \(14 - 7 = 7\)
- \(23 - 14 = 9\)

The differences are \(5, 7, 9\), which are increasing by \(2\). This indicates that the series is not an arithmetic progression (AP), but the differences form an arithmetic progression.

### Step 2: Establish the general term
Let’s denote the \(n^{th}\) term of the series as \(T_n\). The first differences are:
- \(D_1 = 5\)
- \(D_2 = 7\)
- \(D_3 = 9\)

The differences of the differences are constant:
- \(7 - 5 = 2\)
- \(9 - 7 = 2\)

This indicates that the second differences are constant, confirming that the original series can be represented as a quadratic function.

Assuming \(T_n = an^2 + bn + c\), we can use the first few terms to set up equations:
1. \(T_1 = 2 \Rightarrow a(1)^2 + b(1) + c = 2\)
2. \(T_2 = 7 \Rightarrow a(2)^2 + b(2) + c = 7\)
3. \(T_3 = 14 \Rightarrow a(3)^2 + b(3) + c = 14\)

This gives us the system of equations:
1. \(a + b + c = 2\) (1)
2. \(4a + 2b + c = 7\) (2)
3. \(9a + 3b + c = 14\) (3)

### Step 3: Solve the system of equations
Subtract (1) from (2):
\[ (4a + 2b + c) - (a + b + c) = 7 - 2 \]
\[ 3a + b = 5 \quad \text{(4)} \]

Subtract (2) from (3):
\[ (9a + 3b + c) - (4a + 2b + c) = 14 - 7 \]
\[ 5a + b = 7 \quad \text{(5)} \]

Now, subtract (4) from (5):
\[ (5a + b) - (3a + b) = 7 - 5 \]
\[ 2a = 2 \Rightarrow a = 1 \]

Substituting \(a = 1\) into (4):
\[ 3(1) + b = 5 \Rightarrow 3 + b = 5 \Rightarrow b = 2 \]

Substituting \(a = 1\) and \(b = 2\) into (1):
\[ 1 + 2 + c = 2 \Rightarrow 3 + c = 2 \Rightarrow c = -1 \]

Thus, the general term is:
\[ T_n = n^2 + 2n - 1 \]

### Step 4: Find the 99th term
Now we can find the 99th term \(T_{99}\):
\[ T_{99} = 99^2 + 2(99) - 1 \]
\[ = 9801 + 198 - 1 \]
\[ = 9998 \]

### Conclusion
The 99th term of the series is \(9998\).

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