Math Problem Statement

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

Solution

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 ada - d, aa, and a+da + d, where aa is the middle term and dd is the common difference.

Step 2: Use the sum of the three terms.

The sum of the three terms is given as 33: (ad)+a+(a+d)=33(a - d) + a + (a + d) = 33 Simplifying this, we get: 3a=333a = 33 a=11a = 11

Step 3: Use the product of the three terms.

The product of the three terms is given as 1287: (ad)a(a+d)=1287(a - d) \cdot a \cdot (a + d) = 1287 Substitute a=11a = 11: (11d)11(11+d)=1287(11 - d) \cdot 11 \cdot (11 + d) = 1287 Simplify the equation: 11(112d2)=128711 \cdot (11^2 - d^2) = 1287 11(121d2)=128711 \cdot (121 - d^2) = 1287 Divide by 11: 121d2=117121 - d^2 = 117 d2=121117=4d^2 = 121 - 117 = 4 d=±2d = \pm 2

Step 4: Determine the three numbers.

If d=2d = 2, the three numbers are 112=911 - 2 = 9, 1111, and 11+2=1311 + 2 = 13.

If d=2d = -2, the three numbers would be 11+2=1311 + 2 = 13, 1111, and 112=911 - 2 = 9.

So, the three consecutive terms are 99, 1111, and 1313.

Step 5: Find the 7th and 14th terms of the AP.

The general term of the AP is given by: an=a+(n1)da_n = a + (n-1) \cdot d For a=11a = 11 and d=2d = 2:

  • The 7th term is: a7=11+(71)2=11+12=23a_7 = 11 + (7-1) \cdot 2 = 11 + 12 = 23
  • The 14th term is: a14=11+(141)2=11+26=37a_{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.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Progression (AP)
Sum of terms in AP
Product of terms in AP
General term of AP

Formulas

General term of an arithmetic progression: \( a_n = a + (n-1) \cdot d \)

Theorems

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Suitable Grade Level

Grades 8-10