The sum of three consecutive numbers of AP is33 and the product is 1287Then determine the numbers and also find the 7th and 14th terms

Let's solve the problem step by step.

### Step 1: Define the three consecutive terms of the arithmetic progression (AP).
Let the three consecutive terms be \( a - d \), \( a \), and \( a + d \), where \( a \) is the middle term and \( d \) is the common difference.

### Step 2: Use the sum of the three terms.
The sum of the three terms is given as 33:
\[
(a - d) + a + (a + d) = 33
\]
Simplifying this, we get:
\[
3a = 33
\]
\[
a = 11
\]

### Step 3: Use the product of the three terms.
The product of the three terms is given as 1287:
\[
(a - d) \cdot a \cdot (a + d) = 1287
\]
Substitute \( a = 11 \):
\[
(11 - d) \cdot 11 \cdot (11 + d) = 1287
\]
Simplify the equation:
\[
11 \cdot (11^2 - d^2) = 1287
\]
\[
11 \cdot (121 - d^2) = 1287
\]
Divide by 11:
\[
121 - d^2 = 117
\]
\[
d^2 = 121 - 117 = 4
\]
\[
d = \pm 2
\]

### Step 4: Determine the three numbers.
If \( d = 2 \), the three numbers are \( 11 - 2 = 9 \), \( 11 \), and \( 11 + 2 = 13 \).

If \( d = -2 \), the three numbers would be \( 11 + 2 = 13 \), \( 11 \), and \( 11 - 2 = 9 \).

So, the three consecutive terms are \( 9 \), \( 11 \), and \( 13 \).

### Step 5: Find the 7th and 14th terms of the AP.
The general term of the AP is given by:
\[
a_n = a + (n-1) \cdot d
\]
For \( a = 11 \) and \( d = 2 \):
- The 7th term is:
\[
a_7 = 11 + (7-1) \cdot 2 = 11 + 12 = 23
\]
- The 14th term is:
\[
a_{14} = 11 + (14-1) \cdot 2 = 11 + 26 = 37
\]

### Final Answer:
- The three consecutive terms are **9, 11, and 13**.
- The 7th term of the AP is **23**.
- The 14th term of the AP is **37**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the common difference in an arithmetic progression if the sum of three consecutive terms is given?
2. How do you derive the general term of an arithmetic progression?
3. What are the properties of the sum and product of terms in an arithmetic progression?
4. How can you find the nth term of an arithmetic progression if the common difference is known?
5. What is the formula for the sum of the first n terms of an arithmetic progression?

**Tip:** The product of terms in an arithmetic progression is often easier to work with by expressing terms around a middle term, especially when dealing with an odd number of terms.

Learn how to solve the problem of finding three consecutive numbers in an arithmetic progression with a step-by-step solution. This problem covers arithmetic progression concepts such as the sum and product of terms, as well as finding specific terms in the progression. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation