College

Thank you for visiting The first term of an arithmetic progression A P is 16 the last term is 60 and the sum is 330 Find the number of. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!

The first term of an arithmetic progression (A.P.) is 16, the last term is 60, and the sum is 330.

Find the number of terms in the series and the common difference.

Answer :

To solve the problem of finding the number of terms in the arithmetic progression and the common difference, let's follow these steps:

1. Identify the Given Information:
- First term [tex]\( a = 16 \)[/tex]
- Last term [tex]\( l = 60 \)[/tex]
- Sum of the series [tex]\( S_n = 330 \)[/tex]

2. Formula for the Sum of an Arithmetic Progression:
The sum of the first [tex]\( n \)[/tex] terms of an arithmetic progression (A.P.) can be given by:
[tex]\[
S_n = \frac{n}{2} \times (a + l)
\][/tex]
where [tex]\( n \)[/tex] is the number of terms.

3. Use the Sum Formula to Find the Number of Terms:
Plug the given values into the sum formula:
[tex]\[
330 = \frac{n}{2} \times (16 + 60)
\][/tex]
Simplify the equation:
[tex]\[
330 = \frac{n}{2} \times 76
\][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[
660 = n \times 76
\][/tex]
Now, solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{660}{76} \approx 8.6842
\][/tex]

4. Formula for Finding the Common Difference in A.P.:
The formula for the last term in an arithmetic progression is:
[tex]\[
l = a + (n - 1) \times d
\][/tex]
where [tex]\( d \)[/tex] is the common difference.

5. Use the Expanded Formula to Find the Common Difference:
Substitute the known values:
[tex]\[
60 = 16 + (8.6842 - 1) \times d
\][/tex]
Simplify:
[tex]\[
60 = 16 + 7.6842 \times d
\][/tex]
Isolate [tex]\( d \)[/tex] by moving 16 to the other side:
[tex]\[
44 = 7.6842 \times d
\][/tex]
Solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{44}{7.6842} \approx 5.726
\][/tex]

So, the number of terms in the series is approximately 8.68, and the common difference is approximately 5.73.

Thank you for reading the article The first term of an arithmetic progression A P is 16 the last term is 60 and the sum is 330 Find the number of. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!

Rewritten by : Jeany

Other Questions