Sum of First ‘n’ Terms of an Arithmetic Progressions

In many real-life situations, we need to find the total of several terms of such a sequence—for example, total monthly savings, total seats in successive rows of an auditorium, or total distance covered when equal amounts are added each hour. The sum of the first n terms of an Arithmetic Progression helps us find such totals quickly and systematically, without adding every term one by one.

If a, n, l are known → \[S_n=\frac{n}{2}(a+l)\]

If a, n, d are known → \[S_n=\frac{n}{2}\left[2a+(n-1)d\right]\]

\[1+2+\cdots+n=\frac{n(n+1)}{2}\]

\[2+4+\cdots+2n=n(n+1)\]

\[1+3+\cdots+(2n-1)=n^2\]

Question: Let's find the sum of first 100 terms of A.P. 14, 16, 18, ….

Solution:

Step 1: Identify the given values from the A.P. ( \[ 14, 16, 18, \ldots \] )

• First term \[ a = 14 \]
• Common difference \[ d = 16 - 14 = 2 \]
• Number of terms \[ n = 100 \]

Step 2: Use the sum of \[ n \] terms formula ( \[ S_{n} \] )

\[ S_{n} = \frac{n}{2}[2a + (n - 1)d] \]

Step 3: Substitute and evaluate

\[ S_{100} = \frac{100}{2}[2(14) + (100 - 1) \times 2] \]

\[ = 50[28 + 198] \]

\[ = 50 \times 226 = 11{,}300 \]

Final Answer:

\[ \therefore \] The sum of the first 100 terms of the given A.P. is \[ 11{,}300. \]

Question: Find the sum of first n odd natural numbers.

Solution : First \[ n \] natural numbers

\[ 1,\ 3,\ 5,\ 7,\ \ldots,\ (2n-1). \]

\[ a = t_1 = 1 \] and \[ t_n = (2n-1),\quad d = 2 \]

\[ \begin{array}{l|l|l} \textbf{Method I} & \textbf{Method II} & \textbf{Method III} \\[4pt] S_n = \dfrac{n}{2}[t_1+t_n] & S_n = \dfrac{n}{2}[2a+(n-1)d] & 1+3+\ldots\ 2n-1 \\[8pt] = \dfrac{n}{2}[1+(2n-1)] & = \dfrac{n}{2}[2\times 1+(n-1)\times 2] & = (1+2+3+\ldots+2n) \\[4pt] & & \quad - (2+4+6+\ldots+2n) \\[8pt] = \dfrac{n}{2}[1+2n-1] & = \dfrac{n}{2}[2+2n-2] & = \dfrac{2n(2n+1)}{2} - \dfrac{2n(n+1)}{2} \\[8pt] = \dfrac{n}{2}\times 2n & = \dfrac{n}{2}\times 2n & = (2n^2+n) - (n^2+n) \\[8pt] = n^2 & = n^2 & = n^2 \end{array} \]

• An A.P. has the form \[ a,\; a + d,\; a + 2d, \ldots. \]
• The \[ n\text{th} \] term is \[ a_{n} = a + (n - 1)d. \]
• The sum of the first \[ n \] terms is: \[ S_{n} = \frac{n}{2}[2a + (n - 1)d] \]
• If the last term is known: \[ S_{n} = \frac{n}{2}(a + l) \]
• Sum of first \[ n \] natural numbers: \[ \frac{n(n + 1)}{2} \]
• Sum of first \[ n \] even numbers: \[ n(n + 1) \]
• Sum of first \[ n \] odd numbers: \[ n^{2} \]