The best way to answer "How can we solve the sum?" depends entirely on which sum we are talking about. The reference material provides formulas for calculating specific types of sums. Here’s a breakdown of how to approach solving different sums, based on the provided information:
Common Summation Formulas
Here's a summary of the formulas provided, which can be used to solve specific types of sums:
| Type of Sum | Formula |
|---|---|
| Sum of first n natural numbers | n(n + 1)/2 |
| Sum of first n even numbers | n(n + 1) |
| Sum of first n odd numbers | n2 |
| Sum of squares of first n natural numbers | [n(n + 1)(2n + 1)]/6 |
How to Solve Different Types of Sums
Here's how to apply these formulas to solve for the sum:
1. Sum of First n Natural Numbers
Formula: n(n + 1)/2
Example: Find the sum of the first 10 natural numbers (1 + 2 + 3 + ... + 10).
- Here, n = 10.
- Sum = 10(10 + 1)/2 = 10(11)/2 = 110/2 = 55
2. Sum of First n Even Numbers
Formula: n(n + 1)
Example: Find the sum of the first 5 even numbers (2 + 4 + 6 + 8 + 10).
- Here, n = 5.
- Sum = 5(5 + 1) = 5(6) = 30
3. Sum of First n Odd Numbers
Formula: n2
Example: Find the sum of the first 7 odd numbers (1 + 3 + 5 + 7 + 9 + 11 + 13).
- Here, n = 7.
- Sum = 72 = 49
4. Sum of Squares of First n Natural Numbers
Formula: [n(n + 1)(2n + 1)]/6
Example: Find the sum of the squares of the first 4 natural numbers (12 + 22 + 32 + 42).
- Here, n = 4.
- Sum = [4(4 + 1)(2*4 + 1)]/6 = [4(5)(9)]/6 = 180/6 = 30
In General
To solve any sum, you need to:
- Identify the pattern or type of sum. Is it a simple arithmetic series? A geometric series? Does it fit one of the formulas above?
- Determine the values. Find the value of 'n' (the number of terms), the first term, the common difference (if applicable), etc.
- Apply the appropriate formula or method. Use the correct formula to calculate the sum, or use other methods like mathematical induction if the formula is unknown.
