Question 9

यदि किसी A.P. के तीसरे और नौवें पद क्रमशः 4 और -8 हैं, तो इसका कौन-सा पद शून्य होगा?

Given: $a_3 = a + 2d = 4$ $a_9 = a + 8d = -8$

Subtracting the first equation from the second: $(a + 8d) - (a + 2d) = -8 - 4$ $6d = -12$ $d = -2$

Now, substituting $d$ back into the first equation: $a + 2(-2) = 4$ $a - 4 = 4$ $a = 8$

We need to find $n$ such that $a_n = 0$: $a_n = a + (n-1)d$ $0 = 8 + (n-1)(-2)$ $0 = 8 - 2n + 2$ $0 = 10 - 2n$ $2n = 10$ $n = 5$

Thus, the 5th term is zero.

Question 10

किसी A.P. का 17वाँ पद उसके 10वें पद से 7 अधिक है। इसका सार्व अंतर ज्ञात कीजिए।

Given: $a_{17} = a_{10} + 7$

Using the nth term formula: $a + 16d = (a + 9d) + 7$ $a + 16d = a + 9d + 7$ $16d - 9d = 7$ $7d = 7$ $d = 1$

Thus, the common difference $d$ is 1.

Question 11

A.P.: 3, 15, 27, 39, ... का कौन-सा पद उसके 54वें पद से 132 अधिक होगा?

First term $a = 3$
Common difference $d = 15 - 3 = 12$

We need to find $n$ such that $a_n = a_{54} + 132$.

First, find the 54th term: $a_{54} = a + 53d$ $= 3 + 53 \cdot 12$ $= 3 + 636$ $= 639$

Now, we need: $a_n = 639 + 132$ $a_n = 771$

Using the nth term formula: $a_n = a + (n-1)d$ $771 = 3 + (n-1) \cdot 12$ $771 = 3 + 12n - 12$ $771 = 12n - 9$ $780 = 12n$ $n = \frac{780}{12} = 65$

Thus, the 65th term is 132 more than the 54th term.

Question 12

दो समांतर श्रेणियों का सार्व अंतर समान है। यदि इनके 100वें पदों का अंतर 100 है, तो इनके 1000वें पदों का अंतर क्या होगा?

Let the first term and common difference of the first A.P. be $a_1$ and $d$, and for the second A.P. be $a_2$ and $d$.

Given: $a_1 + 99d - (a_2 + 99d) = 100$ $a_1 - a_2 = 100$

For the 1000th term: $a_1 + 999d - (a_2 + 999d) = a_1 - a_2$ $a_1 - a_2 = 100$

Thus, the difference between the 1000th terms will also be 100.

If you need further clarifications or detailed explanations, please let me know. Here are some additional questions for practice:

1. How to find the common difference if the sum of certain terms is known?
2. How to determine the first term of an A.P. if the common difference and certain terms are known?
3. What are the key properties of arithmetic progressions?
4. How to find the sum of the first $n$ terms of an A.P.?
5. How to handle arithmetic progressions with negative common differences?

Tip: Practice with different types of problems involving A.P. to strengthen your understanding and problem-solving skills.