Understanding the Problem: Finding the Fourth Term of an AP

The question provides the sum of the first \(n\) terms of an arithmetic progression (AP) as \(S_n = n(n + 1)\). We need to find the value of the fourth term in this AP.

In an arithmetic progression, each term after the first is obtained by adding a constant value, called the common difference, to the preceding term.

Calculating Terms from the Sum Formula

We can find individual terms of an AP if we know the sum of terms formula. The first term \(a_1\) is simply the sum of the first term, \(S_1\). The nth term \(a_n\) can be found by subtracting the sum of the first \(n-1\) terms from the sum of the first \(n\) terms, i.e., \(a_n = S_n - S_{n-1}\) for \(n > 1\).

Let's calculate the first few sums using the given formula \(S_n = n(n + 1)\):

For \(n=1\), \(S_1 = 1(1 + 1) = 1 \times 2 = 2\).

For \(n=2\), \(S_2 = 2(2 + 1) = 2 \times 3 = 6\).

For \(n=3\), \(S_3 = 3(3 + 1) = 3 \times 4 = 12\).

For \(n=4\), \(S_4 = 4(4 + 1) = 4 \times 5 = 20\).

Now we can find the terms of the AP:

The first term \(a_1\) is \(S_1\).
\(a_1 = S_1 = 2\).

The second term \(a_2\) is \(S_2 - S_1\).
\(a_2 = S_2 - S_1 = 6 - 2 = 4\).

The third term \(a_3\) is \(S_3 - S_2\).
\(a_3 = S_3 - S_2 = 12 - 6 = 6\).

The fourth term \(a_4\) is \(S_4 - S_3\).
\(a_4 = S_4 - S_3 = 20 - 12 = 8\).

Verifying the AP and Finding the Common Difference

The first few terms are 2, 4, 6, ... Let's check if this is an AP by finding the common difference (\(d\)).

Difference between the second and first term: \(a_2 - a_1 = 4 - 2 = 2\).

Difference between the third and second term: \(a_3 - a_2 = 6 - 4 = 2\).

Since the difference is constant (\(d=2\)), this is indeed an AP with the first term \(a_1 = 2\) and common difference \(d = 2\).

Calculating the Fourth Term using the AP Formula

Alternatively, once we have \(a_1\) and \(d\), we can find any term using the formula for the nth term of an AP: \(a_n = a_1 + (n - 1)d\).

For the fourth term (\(n=4\)):

\(a_4 = a_1 + (4 - 1)d\)

\(a_4 = a_1 + 3d\)

Substituting the values \(a_1 = 2\) and \(d = 2\):

\(a_4 = 2 + 3(2)\)

\(a_4 = 2 + 6\)

\(a_4 = 8\)

Both methods confirm that the fourth term of the AP is 8.

The terms of the AP are 2, 4, 6, 8, ...

Conclusion on the Fourth Term

Based on our calculations using both the sum formula difference method and the AP term formula, the fourth term of the AP is 8.

Revision Table: AP Concepts

Additional Information: Deriving AP Properties

Let's see how the given sum formula \(S_n = n(n+1)\) relates to the standard AP formulas.

We found \(a_1 = 2\) and \(d = 2\).

Let's substitute these into the standard sum formula \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\):

\(S_n = \frac{n}{2}(2(2) + (n-1)2)\)

\(S_n = \frac{n}{2}(4 + 2n - 2)\)

\(S_n = \frac{n}{2}(2n + 2)\)

\(S_n = \frac{n}{2} \times 2(n + 1)\)

\(S_n = n(n + 1)\)

This confirms that the given sum formula corresponds to an AP with \(a_1 = 2\) and \(d = 2\). The terms are 2, 4, 6, 8, 10, ...