https://file.aiquickdraw.com/mathbot/file/1726966686626pdc0gwy6.jpg

Let's solve the problems from the image, starting with **question 10**.

---

### **10. The 8th term of an AP is 31 and its 15th term exceeds its 11th term by 16. Find that AP.**

Let the first term of the arithmetic progression (AP) be \( a \), and the common difference be \( d \).

The general formula for the \( n \)-th term of an AP is given by:

\[
T_n = a + (n-1) d
\]

#### Given Information:

- The 8th term \( T_8 = 31 \)
\[
a + 7d = 31 \tag{1}
\]
- The difference between the 15th and 11th terms is 16:
\[
T_{15} - T_{11} = 16
\]
Using the formula for the terms:
\[
(a + 14d) - (a + 10d) = 16
\]
Simplifying:
\[
4d = 16 \implies d = 4 \tag{2}
\]

#### Substitute \( d = 4 \) in equation (1):
\[
a + 7(4) = 31
\]
\[
a + 28 = 31 \implies a = 3 \tag{3}
\]

Thus, the first term \( a = 3 \) and the common difference \( d = 4 \).

Therefore, the AP is:
\[
3, 7, 11, 15, 19, \dots
\]

---

### **11. How many two-digit positive integers are divisible by 7?**

The smallest two-digit number divisible by 7 is \( 14 \times 7 = 98 \), and the largest is \( 10 \times 7 = 70 \).

The two-digit numbers divisible by 7 form an arithmetic progression (AP) with:

- First term \( a = 14 \times 7 = 98 \)
- Common difference \( d = 7 \)
- Last term \( l = 70 \)

The number of terms \( n \) in the AP is found using the formula for the last term:
\[
l = a + (n-1) d
\]
Substituting values:
\[
98 = 7 + (10 - 1) d

Explore solutions to arithmetic progression problems from question 10 onwards. Learn how to find specific terms of APs, calculate the number of terms divisible by a given number, and more. Perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation