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Find the Sum of the Odd numbers between 0 and 50
To find the sum of odd numbers between 0 and 50, you can use the formula for the sum of an arithmetic series:
S = (n/2) * (first term + last term)
In this case, since we’re summing odd numbers, the sequence is 1, 3, 5, 7, …, 49. We can see that this forms an arithmetic sequence with a common difference of 2.
The first term, a₁, is 1.The last term, aₙ, is 49.
The number of terms can be calculated using the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n – 1) * d
49 = 1 + (n – 1) * 2
49 = 1 + 2n – 2
49 = 2n – 1
50 = 2n
n = 25
Now we have the number of terms (n) as 25. We can plug these values into the formula for the sum of an arithmetic series:
S = (n/2) * (a₁ + aₙ)
S = (25/2) * (1 + 49)
S = (25/2) * 50
S = (25 * 50)/2
S = 625
So, the sum of odd numbers between 0 and 50 is 625.
What is Arithmetic Progression?
Arithmetic progression (AP), also known as an arithmetic sequence, is a special type of sequence where the difference between consecutive terms remains constant. In simpler terms, imagine a ladder where each rung is spaced evenly apart. The distance between each rung represents the common difference in an AP.
Here are some key characteristics of an AP:
- Constant difference: Each term is obtained by adding (or subtracting) the same value, called the common difference (d), to the previous term.
- Formula for nth term: You can find any term (n th term) in the sequence using the formula a_n = a_1 + (n – 1)d, where a_1 is the first term and d is the common difference.
- Sum of an AP: The sum of all terms in a finite AP can be calculated using the formula S_n = n/2 (2a_1 + (n – 1)d), where n is the number of terms, a_1 is the first term, and d is the common difference.
Here are some examples of APs:
- 2, 5, 8, 11, 14, … (common difference = 3)
- 10, 7, 4, 1, -2, … (common difference = -3)
- 3, 6.3, 9.6, 12.9, 16.2, … (common difference = 3.3)
Arithmetic progressions have numerous applications in various fields, including:
- Finance: Calculating compound interest or future values of investments.
- Physics: Describing uniform motion with constant acceleration.
- Statistics: Analyzing frequency distributions of data.
- Computer science: Generating numerical sequences for algorithms.
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