IF `S_n={(n(n+1))/2}^2` when `S_n` is the sum of first n terms of the series then its rth term is

Options

A. r
B. `r^2`
C. `r^3`
D. `(r^2(r+1)^2)/4`

Answer

To find the rth term of the series given that \( S_n = \left( \frac{n(n+1)}{2} \right)^2 \), we can follow these steps:

### Step 1: Understand the formula for \( S_n \)
The expression \( S_n = \left( \frac{n(n+1)}{2} \right)^2 \) represents the sum of the first \( n \) terms of a series. This is known to be the sum of the cubes of the first \( n \) natural numbers.

### Step 2: Identify the relationship between \( S_n \) and the rth term
The rth term of the series, denoted as \( T_r \), can be found using the relationship:
\[ T_r = S_r - S_{r-1} \]
This means that the rth term is equal to the sum of the first \( r \) terms minus the sum of the first \( r-1 \) terms.

### Step 3: Calculate \( S_r \) and \( S_{r-1} \)
Using the formula for \( S_n \):
- For \( S_r \):
\[ S_r = \left( \frac{r(r+1)}{2} \right)^2 \]
- For \( S_{r-1} \):
\[ S_{r-1} = \left( \frac{(r-1)r}{2} \right)^2 \]

### Step 4: Substitute into the formula for \( T_r \)
Now, substitute \( S_r \) and \( S_{r-1} \) into the equation for \( T_r \):
\[ T_r = S_r - S_{r-1} = \left( \frac{r(r+1)}{2} \right)^2 - \left( \frac{(r-1)r}{2} \right)^2 \]

### Step 5: Simplify the expression
We can simplify \( T_r \):
\[ T_r = \left( \frac{r(r+1)}{2} \right)^2 - \left( \frac{(r-1)r}{2} \right)^2 \]
This can be factored using the difference of squares:
\[ T_r = \left( \frac{1}{4} \right) \left( r^2(r+1)^2 - (r-1)^2r^2 \right) \]
\[ = \frac{1}{4} r^2 \left( (r+1)^2 - (r-1)^2 \right) \]
Now, calculate \( (r+1)^2 - (r-1)^2 \):
\[ (r+1)^2 - (r-1)^2 = (r^2 + 2r + 1) - (r^2 - 2r + 1) = 4r \]
Thus,
\[ T_r = \frac{1}{4} r^2 (4r) = r^3 \]

### Final Result
The rth term of the series is:
\[ T_r = r^3 \]