Solving AP Problems: Finding the Sum of the First 3 Terms of an Arithmetic Progression

Understanding and solving problems involving arithmetic progressions (AP) can significantly enhance mathematical reasoning skills. This article delves into the method of finding the sum of the first three terms when given a specific term and the sum of the first 21 terms. Follow the step-by-step solution to master the technique.

Introduction to Arithmetic Progression (AP)

Arithmetic Progression, or AP, is a sequence of numbers where the difference between any two successive members is a constant. This difference is known as the common difference, denoted by (d).

Given Information and Formulas

The 21st term, (a_{21}) (frac{11}{2}) The sum of the first 21 terms, (S_{21}) (frac{189}{2})

The formulas we will use:

The (n)-th term of an AP: (a_n a (n-1)d) The sum of the first (n) terms of an AP: (S_n frac{n}{2} [2a (n-1)d])

Step-by-Step Solution

Step 1: Express the 21st term using the formula.

(a_{21} a 20d frac{11}{2})

Step 2: Use the sum formula to express the sum of the first 21 terms.

(S_{21} frac{21}{2} [2a 20d] frac{189}{2})

Step 3: Simplify the sum equation to eliminate the fractions.

Multiplying both sides by 2:

21 [2a 20d] 189

Dividing both sides by 21:

[2a 20d] 9

Step 4: Set up a system of equations.

Equation 1: (a 20d frac{11}{2})

Equation 2: (2a 20d 9)

Step 5: Solve the equations.

From Equation 2, express (a) in terms of (d):

2a 9 - 20d implies (a frac{9 - 20d}{2})

Substitute (a) back into Equation 1:

(frac{9 - 20d}{2} 20d frac{11}{2})

Multiplying the entire equation by 2 to eliminate the fraction:

(9 - 20d 40d 11)

Combining like terms:

(9 20d 11)

Solving for (d):

(20d 2) implies (d frac{1}{10})

Step 6: Find (a).

(a frac{9 - 20 cdot frac{1}{10}}{2} frac{9 - 2}{2} frac{7}{2})

Step 7: Sum of the first 3 terms.

The first three terms of the AP are:

First term: (a frac{7}{2}) Second term: (a d frac{7}{2} frac{1}{10} frac{35}{10} frac{1}{10} frac{36}{10} frac{18}{5}) Third term: (a 2d frac{7}{2} 2 cdot frac{1}{10} frac{7}{2} frac{2}{10} frac{7}{2} frac{1}{5} frac{35}{10} frac{2}{10} frac{37}{10})

To add these fractions, we need a common denominator. The least common multiple of 2, 5, and 10 is 10:

(frac{35}{10} frac{36}{10} frac{37}{10} frac{35 36 37}{10} frac{108}{10} frac{54}{5})

Final Answer

The sum of the first 3 terms of the AP is (frac{54}{5}) or (10 frac{4}{5}).

Conclusion

Mastering the technique for solving AP problems, such as finding the sum of the first 3 terms given the 21st term and the sum of the first 21 terms, is crucial for honing mathematical skills. This step-by-step guide walks through each necessary calculation and helps ensure a solid understanding of the underlying concepts.