Math Problem Statement

In an AP the 16th Term is three time the 5th Term and the 12th Term is 100

What is the sIn an AP the 16th Term is three time the 5th Term and the 12th Term is 100. What is the sum of the first twenty terms?  	 

. (marks)

Solution

We are given the following information about an Arithmetic Progression (AP):

  1. The 16th term is three times the 5th term.
  2. The 12th term is 100.
  3. We need to find the sum of the first 20 terms of this AP.

Step 1: Formula for the nth term of an AP

The general formula for the nth term of an AP is: Tn=a+(n1)dT_n = a + (n - 1) d where:

  • TnT_n is the nth term,
  • aa is the first term,
  • dd is the common difference.

Step 2: Expressing the 16th term and 5th term

Using the formula for the nth term:

  • The 16th term T16=a+15dT_{16} = a + 15d,
  • The 5th term T5=a+4dT_5 = a + 4d.

From the problem statement, we know that the 16th term is three times the 5th term: a+15d=3(a+4d)a + 15d = 3(a + 4d) Simplifying this equation: a+15d=3a+12da + 15d = 3a + 12d 15d12d=3aa15d - 12d = 3a - a 3d=2a3d = 2a a=3d2a = \frac{3d}{2} (Equation 1)\quad \text{(Equation 1)}

Step 3: Using the information about the 12th term

The 12th term is given as 100: T12=a+11d=100T_{12} = a + 11d = 100 Substitute a=3d2a = \frac{3d}{2} into this equation: 3d2+11d=100\frac{3d}{2} + 11d = 100 Multiply through by 2 to eliminate the fraction: 3d+22d=2003d + 22d = 200 25d=20025d = 200 d=8d = 8

Step 4: Finding the first term aa

Substitute d=8d = 8 into a=3d2a = \frac{3d}{2}: a=3×82=12a = \frac{3 \times 8}{2} = 12

Step 5: Finding the sum of the first 20 terms

The sum of the first nn terms of an AP is given by: Sn=n2×(2a+(n1)d)S_n = \frac{n}{2} \times (2a + (n-1)d) Substitute n=20n = 20, a=12a = 12, and d=8d = 8: S20=202×(2×12+19×8)S_{20} = \frac{20}{2} \times (2 \times 12 + 19 \times 8) S20=10×(24+152)S_{20} = 10 \times (24 + 152) S20=10×176S_{20} = 10 \times 176 S20=1760S_{20} = 1760

Final Answer:

The sum of the first 20 terms of the AP is 1760.

Would you like further clarification or details on any step?


Here are 5 related questions to expand on this:

  1. How would the formula change if we needed to find the sum of the first 30 terms?
  2. What is the common difference in an arithmetic progression if the sum of the first 10 terms is 145?
  3. How do you find the nth term in a geometric progression?
  4. How can you determine the common difference in an AP if only the first and last terms are known?
  5. What is the sum of the first 15 terms of an AP where the 10th term is 50 and the common difference is 5?

Tip: When solving AP problems, always start by writing the nth term formula and substitute the given values step by step.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Progression
Algebra

Formulas

nth term of an AP: T_n = a + (n - 1) d
Sum of the first n terms of an AP: S_n = (n / 2) * (2a + (n - 1) d)

Theorems

Arithmetic Progression Formula

Suitable Grade Level

Grades 9-10