Finding the Fifth Term of an AP from the Sum Formula

The question asks us to find the fifth term of an Arithmetic Progression (AP) given the formula for the sum of its first n terms. The sum of the first n terms is given by $S_n = n^2 - 2n$.

To find any term of an AP, say the n-th term ($a_n$), when the sum of n terms ($S_n$) is known, we can use the relationship:

$a_n = S_n - S_{n-1}$

where $S_{n-1}$ is the sum of the first (n-1) terms. This formula is valid for $n > 1$. For the first term ($a_1$), it is simply equal to $S_1$.

Step-by-Step Calculation of the n-th Term ($a_n$)

We are given the formula for $S_n$:

$S_n = n^2 - 2n$

Now, we need to find the formula for $S_{n-1}$. We substitute $(n-1)$ for $n$ in the $S_n$ formula:

$S_{n-1} = (n-1)^2 - 2(n-1)$

Let's expand and simplify this expression:

$S_{n-1} = (n^2 - 2n + 1) - (2n - 2)$

$S_{n-1} = n^2 - 2n + 1 - 2n + 2$

$S_{n-1} = n^2 - 4n + 3$

Now, we can find the formula for the n-th term, $a_n$, using $a_n = S_n - S_{n-1}$:

$a_n = (n^2 - 2n) - (n^2 - 4n + 3)$

$a_n = n^2 - 2n - n^2 + 4n - 3$

$a_n = (n^2 - n^2) + (-2n + 4n) - 3$

$a_n = 0 + 2n - 3$

$a_n = 2n - 3$

This formula gives us the n-th term of the AP.

Calculating the Fifth Term ($a_5$)

We need to find the fifth term, which means we need to find $a_5$. We substitute $n=5$ into the formula for $a_n$:

$a_5 = 2(5) - 3$

$a_5 = 10 - 3$

$a_5 = 7$

So, the fifth term of the AP is 7.

Let's also check the first term using $a_1 = S_1$.

$S_1 = (1)^2 - 2(1) = 1 - 2 = -1$

$a_1 = -1$

Using the formula $a_n = 2n - 3$ for $n=1$:

$a_1 = 2(1) - 3 = 2 - 3 = -1$

The formulas match for $n=1$.

Summary of Terms

Using the formula $a_n = 2n - 3$, we can find the first few terms:

• First term ($a_1$): $2(1) - 3 = -1$
• Second term ($a_2$): $2(2) - 3 = 4 - 3 = 1$
• Third term ($a_3$): $2(3) - 3 = 6 - 3 = 3$
• Fourth term ($a_4$): $2(4) - 3 = 8 - 3 = 5$
• Fifth term ($a_5$): $2(5) - 3 = 10 - 3 = 7$

The sequence starts with -1, 1, 3, 5, 7, ...

The common difference of this AP is $1 - (-1) = 2$ or $3 - 1 = 2$.

Revision Table: Key AP Formulas

Additional Information: Properties of AP

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$.

• The general form of an AP is $a_1, a_1+d, a_1+2d, a_1+3d, \dots$
• The common difference $d$ can be found by subtracting any term from its succeeding term, e.g., $d = a_2 - a_1 = a_3 - a_2$, and so on.
• If the formula for $S_n$ is a quadratic expression in $n$ of the form $An^2 + Bn$, then the sequence is an AP. The common difference $d$ is equal to $2A$, and the first term $a_1$ is equal to $A+B$.

In our case, $S_n = n^2 - 2n$. This is in the form $An^2 + Bn$ with $A=1$ and $B=-2$.

• Common difference $d = 2A = 2(1) = 2$.
• First term $a_1 = A + B = 1 + (-2) = -1$.

Using $a_1 = -1$ and $d = 2$, the n-th term is $a_n = a_1 + (n-1)d = -1 + (n-1)2 = -1 + 2n - 2 = 2n - 3$. This confirms the formula for $a_n$ derived earlier using $S_n - S_{n-1}$.