The sum of the first terms of an AP is a widely used formula in mathematics to find the total value of terms in a sequence without manually adding each term. This formula is applicable in solving problems across finance, physics, and engineering.
Formula for the Sum of First n Terms
The sum of the first terms of an Arithmetic Progression (AP) is denoted by . There are two primary formulas:
- If the first term () and the common difference () are known:
- If the first term () and the last term () are known:
Here:
- = the sum of the first terms,
- = the first term of the AP,
- = the common difference between consecutive terms,
- = the total number of terms,
- = the last term of the AP.
Derivation of the Formula
Using the Sequence Structure
Consider an AP: .
The sum of the first terms can be expressed as:
.
Writing this sum in reverse order:
.
Add these two equations:
.
Divide both sides by :
.
Examples with Step-by-Step Solutions
Example 1: Find the sum of the first 15 terms of the AP .
Solution:
- Identify the first term and the common difference:
. - Use the formula for the sum of the first terms:
. - Substitute the values of , , and :
. - Simplify the expression:
.
Answer: The sum of the first 15 terms is .
Example 2: The sum of the first 10 terms of an AP is , and the first term is . Find the common difference.
Solution:
- Use the formula for the sum of the first terms:
. - Substitute the known values:
. - Simplify the equation:
. - Solve for :
.
Answer: The common difference is .
Example 3: Find the sum of all multiples of between and .
Solution:
- Form the AP:
The multiples of between and are:
.
Here, , , and . - Find the number of terms ():
Use the formula for the nth term:
.
Substituting , , and :
. - Use the formula for the sum of the first terms:
.
Substituting , , and :
.
Answer: The sum of all multiples of between and is .
Example 4: Find the sum of the first 22 terms of the AP: .
Solution:
- Identify the values:
, , . - Use the formula for the sum of terms:
- Substitute the values:
- Simplify the expression:
Answer:
Example 5: If the sum of the first 14 terms of an AP is , and its first term is , find the 20th term.
Solution:
- Identify the values:
, , . - Use the formula for the sum of terms:
- Substitute the values for the 14th term sum:
- Simplify the equation:
- Find the 20th term using the nth term formula:
Answer:
Example 6: Find the sum of the first 24 terms of the list of numbers whose nth term is given by .
Solution:
- Expand the sequence:
, , , - Identify the values:
, , . - Use the formula for the sum of terms:
- Substitute the values:
- Simplify the expression:
Answer:
Example 7: A manufacturer produces 600 TV sets in the third year and 700 in the seventh year. Assuming production increases uniformly, find:
- The production in the first year,
- The production in the 10th year,
- The total production in the first 7 years.
Solution:
- Since production increases uniformly, it forms an AP. Let the first year production be and the common difference be .
From the data:
Subtract the first equation from the second:
Substitute into :
Answer: Production in the first year: . - Find the production in the 10th year:
Answer: Production in the 10th year: . - Find the total production in the first 7 years:
Answer: Total production in the first 7 years:
Practice Problems
- Find the sum of the first terms of the AP .
Answer: . - The 8th term of an AP is , and the sum of the first terms is . Find the first term.
Answer: . - Find the sum of all two-digit numbers divisible by .
Answer: .
FAQs
The sum is derived by pairing terms of the sequence in reverse order, resulting in a simplified formula.
It helps in solving problems involving large sequences without manually adding all terms.
Related Topics
- Solution of a Quadratic Equation by Factorisation
- Differentiation Formulas
- Real Numbers
- Similarity of Triangles
- Introduction to Quadratic Equations
- Pair of Linear Equations in Two Variables
- Maths FAQs
- nth Term of an AP
- Ordinal Numbers
- Trigonometry Formulas
- Pythagoras Theorem
- Percentage
- Introduction to Polynomials
- Probability
- Sets
Get Social