3. (a) If the arithmetic mean between x and 2x is 3, find the value of x.

(b) If the arithmetic mean between y-1 and 2y is 7, find the value of y.

4. (a) In an AP, 4th mean is 12 and the ratio of first mean to the 4th mean is 3. Find the first mean.

(b) Write the number of terms which is equal to 5th mean.

5. (a) In an AP, 10th and 12th terms are 100 and 120 respectively, find the 11th term.

(b) If 10, x, 20 are in AP, find x.

6. (a) Find the value of x and y if 10, x, y, 25 are in AP.

(b) If-13, p, q, r, 7 are in AP, find the values of p, q and r.

7. (a) 1+n If /, m and n are in AP, show that m =.

(b) If the arithmetic mean between two number is 50 and the second number is 60, find the first number.

(c) Two numbers are in the ratio of 2:3. If their AM is 20, find the numbers.

8. (a) Insert two arithmetic means between 4 and 22.

(b) Insert two arithmetic means between 100 and 10.

9. (a) Insert 5 arithmetic means between 10 and 70.

(b) Insert 6 arithmetic means between 11 and 39.

10. (a) There are 'n' arithmetic means between 15 and 45. If the third mean is 30, find the value of n. Also, find the ratio of 3rd mean to fifth mean.

(b) Some arithmetic means are inserted between 9 and 33. If the third mean is 21, find the number of means and the values of the remaining means.

11. (a) If 'n' arithmetic means are inserted between 5 and 35. The ratio of second and last mean is 1:4, find the value of n.

(b) There are 6 AM's between a and b. If the 2nd mean and 5th means are 11 and 23 respectively, then find the values of a and b.

Let's tackle each of these problems step by step.

### 3. (a) If the arithmetic mean between \( x \) and \( 2x \) is 3, find the value of \( x \).

The arithmetic mean between two numbers \( a \) and \( b \) is given by:
\[
\text{Arithmetic Mean} = \frac{a + b}{2}
\]
For this problem:
\[
\frac{x + 2x}{2} = 3
\]
\[
\frac{3x}{2} = 3
\]
Multiply both sides by 2:
\[
3x = 6
\]
Divide by 3:
\[
x = 2
\]

### 3. (b) If the arithmetic mean between \( y-1 \) and \( 2y \) is 7, find the value of \( y \).

The arithmetic mean here is:
\[
\frac{(y - 1) + 2y}{2} = 7
\]
Simplify the expression:
\[
\frac{3y - 1}{2} = 7
\]
Multiply both sides by 2:
\[
3y - 1 = 14
\]
Add 1 to both sides:
\[
3y = 15
\]
Divide by 3:
\[
y = 5
\]

### 4. (a) In an AP, the 4th mean is 12 and the ratio of the first mean to the 4th mean is 3. Find the first mean.

Let's denote the first mean by \( a \) and the common difference by \( d \).

The 4th mean is the fourth term, which is:
\[
a + 3d = 12
\]
The problem states the ratio of the first mean to the 4th mean is 3:
\[
\frac{a}{a + 3d} = 3
\]
This implies:
\[
a = 3(a + 3d)
\]
Substitute \( a + 3d = 12 \) into the equation:
\[
a = 3 \times 12 = 36
\]

### 4. (b) Write the number of terms which is equal to the 5th mean.

The 5th mean corresponds to the 5th term in the sequence:
\[
a + 4d
\]
Since no specific terms were given, it's general. The number of terms is 5.

### 5. (a) In an AP, the 10th and 12th terms are 100 and 120 respectively. Find the 11th term.

Let's denote the first term by \( a \) and the common difference by \( d \).

The 10th term:
\[
a + 9d = 100
\]
The 12th term:
\[
a + 11d = 120
\]
Subtract the first equation from the second:
\[
(11d - 9d) = 120 - 100
\]
\[
2d = 20 \Rightarrow d = 10
\]
Substitute \( d = 10 \) into the equation for the 10th term:
\[
a + 9(10) = 100 \Rightarrow a + 90 = 100 \Rightarrow a = 10
\]
The 11th term is:
\[
a + 10d = 10 + 10(10) = 110
\]

### 5. (b) If 10, \( x \), 20 are in AP, find \( x \).

For three terms in AP, the middle term \( x \) is the average of the other two:
\[
x = \frac{10 + 20}{2} = \frac{30}{2} = 15
\]

Would you like detailed explanations or have any questions on any of these steps?

Here are 5 related questions:

1. If the arithmetic mean between \( 3x \) and \( 5x \) is 16, what is the value of \( x \)?
2. Find the value of \( z \) if \( z-2, z+4, z+8 \) are in AP.
3. Calculate the 7th term of an AP if the 3rd term is 15 and the common difference is 4.
4. Determine the first term of an AP if the sum of the first 5 terms is 40 and the 3rd term is 10.
5. Find the common difference of an AP where the sum of the first 4 terms is 40, and the first term is 5.

**Tip:** In any arithmetic progression, knowing the first term and the common difference is key to finding any term in the sequence.

Explore arithmetic progression (AP) problems including finding values of x and y based on arithmetic means. Solve for the first mean and other terms in AP sequences. Suitable for Grades 7-9 students learning about arithmetic progression and arithmetic mean.

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