https://r2.erweima.ai/mathbot/image/1720000932489w4sekbxn.jpg

Solve all these problems including concepts if needed 

### Question 4
**A.P.: 3, 8, 13, 18, ... का कौन सा पद 78 है?**

Given A.P.: 3, 8, 13, 18, ...

First term \(a = 3\)  
Common difference \(d = 8 - 3 = 5\)  

We need to find \(n\) for which \(a_n = 78\).

Using the nth term formula of an A.P.:
\[ a_n = a + (n-1)d \]
\[ 78 = 3 + (n-1) \cdot 5 \]
\[ 78 = 3 + 5n - 5 \]
\[ 78 = 5n - 2 \]
\[ 80 = 5n \]
\[ n = \frac{80}{5} = 16 \]

Thus, the 16th term is 78.

### Question 5
**निम्नलिखित समांतर श्रेणियों में से प्रत्येक श्रेणी में कितने पद हैं?**

(i) \( 7, 13, 19, \ldots, 205 \)

First term \(a = 7\)  
Common difference \(d = 13 - 7 = 6\)  
Last term \(a_n = 205\)

Using the nth term formula:
\[ a_n = a + (n-1)d \]
\[ 205 = 7 + (n-1) \cdot 6 \]
\[ 205 = 7 + 6n - 6 \]
\[ 205 = 6n + 1 \]
\[ 204 = 6n \]
\[ n = \frac{204}{6} = 34 \]

So, there are 34 terms.

(ii) \( 18, 15\frac{1}{2}, 13, \ldots, -47 \)

First term \(a = 18\)  
Common difference \(d = 15\frac{1}{2} - 18 = -2.5\)  
Last term \(a_n = -47\)

Using the nth term formula:
\[ a_n = a + (n-1)d \]
\[ -47 = 18 + (n-1)(-2.5) \]
\[ -47 = 18 - 2.5n + 2.5 \]
\[ -47 = 20.5 - 2.5n \]
\[ -67.5 = -2.5n \]
\[ n = \frac{67.5}{2.5} = 27 \]

So, there are 27 terms.

### Question 6
**क्या A.P. 11, 8, 5, 2, ... का एक पद -150 है? क्यों?**

First term \(a = 11\)  
Common difference \(d = 8 - 11 = -3\)  

We need to find if there is any term \(a_n\) such that \(a_n = -150\).

Using the nth term formula:
\[ a_n = a + (n-1)d \]
\[ -150 = 11 + (n-1)(-3) \]
\[ -150 = 11 - 3n + 3 \]
\[ -150 = 14 - 3n \]
\[ -164 = -3n \]
\[ n = \frac{164}{3} \approx 54.67 \]

Since \(n\) is not an integer, \(-150\) is not a term in the given A.P.

### Question 7
**उस A.P. का 31वाँ पद ज्ञात कीजिए, जिसका 11वाँ पद 38 है और 16वाँ पद 73 है।**

Given:
\[ a + 10d = 38 \]
\[ a + 15d = 73 \]

Subtracting the first equation from the second:
\[ (a + 15d) - (a + 10d) = 73 - 38 \]
\[ 5d = 35 \]
\[ d = 7 \]

Now, substituting \(d\) back into the first equation:
\[ a + 10 \cdot 7 = 38 \]
\[ a + 70 = 38 \]
\[ a = 38 - 70 = -32 \]

To find the 31st term:
\[ a + 30d = -32 + 30 \cdot 7 \]
\[ = -32 + 210 \]
\[ = 178 \]

Thus, the 31st term is 178.

### Question 8
**एक A.P. में 50 पद हैं, जिसका तीसरा पद 12 है और अंतिम पद 106 है। इसका 29वाँ पद ज्ञात कीजिए।**

Given:
\[ a_3 = a + 2d = 12 \]
\[ a_{50} = a + 49d = 106 \]

From the first equation:
\[ a + 2d = 12 \]
\[ a = 12 - 2d \]

Substitute \(a\) into the second equation:
\[ (12 - 2d) + 49d = 106 \]
\[ 12 + 47d = 106 \]
\[ 47d = 94 \]
\[ d = 2 \]

Now, substituting \(d\) back into \(a\):
\[ a = 12 - 2 \cdot 2 \]
\[ a = 12 - 4 \]
\[ a = 8 \]

To find the 29th term:
\[ a_{29} = a + 28d \]
\[ = 8 + 28 \cdot 2 \]
\[ = 8 + 56 \]
\[ = 64 \]

Thus, the 29th term is 64.

If you need further explanations or more detailed steps for these problems, please let me know. Here are some questions for you to consider:

1. How to derive the general formula for the nth term of an A.P.?
2. What is the process to find the number of terms in a given A.P.?
3. How to determine if a specific value is a term in an A.P.?
4. How to solve a problem involving multiple terms of an A.P.?
5. What are the practical applications of A.P. in real life?

**Tip:** When working with arithmetic progressions, always check the common difference and first term carefully, as these form the basis of all calculations.

Explore how to solve arithmetic progression (A.P.) problems step-by-step. Learn to find specific terms in A.P., derive the nth term formula, and apply it to real-life examples. Suitable for high school students and those preparing for mathematical challenges.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation