sum of first n terms of an AP derivative
Answers
Answer:
In this sequence, the sum of numbers can be represented as such:
Sum = 1+2+3+4+5+6….+97+98+99+100
Even when the order is reversed, the sum does not change:
Sum = 100+99+98+97….6+5+4+3+2+1
Hence, when both equations 1 and 2 are added, we get:
2 x Sum = (100+1) + (99+2) + (98+3) + (97+4) + … (4+97) + (3+98) + (2+99) + (1+100)
2 x Sum = 101+ 101 + 101 + 101 + … (4+97) + (3+98) + (2+99) + (1+100)
2 x Sum = 101 (100 terms)
2Sum = 101(100)
Sum =
sum of n terms
Sum = 5,050
This method can be used to calculate the sum of natural numbers like 1000, 10,000 or even 100,000. Hence, knowing the last term in the sequence, this method can be used to derive the formula needed to figure out the nth term in any given sequence.
If we express the first term in the academic progression as a, the common difference between each consecutive term as d, and the last term as l.
Sum = a + (a + d) + (a + 2d) + (a + 3d)…+ (l – 3d) + (l – 2d) + (l – d) + l
Where l = a + (n – 1)d
In reverse order, the sum remains the same:
Sum = l + (l – d) + (l – 2d) + (l – 3d) + … (a + 3d) + (a + 2d) + (a + d) + a
Adding equations a and b, we get:
2 x Sum = (a + l) + [(a +d) + (l – d)] … + [(l – d) + (a + d)] + (l + a)]
2 x sum = (a + l) + (a + l)…+ (a + l) + (a + l)
2 x sum = n x (a + l)
sum =
formula for sum of n terms
Substituting l with the previous equation above (where l = a + (n – 1)d):
Sum of n terms of AP =
sum of n terms of ap
Arithmetic Progression
In an arithmetic progression, not all numbers in the sequence may be known or given. To find the nth term, the equation: a + (n – 1)d is used.
Examples
Let’s take the first sequence as an example: 4, 6, 8, 10, 12,…
The table below is used to visualize the components needed to fulfill the arithmetic progression:
Terms n (n-1) d
4 1 0 –
6 2 1 2
8 3 2 2
10 4 3 2
12 5 4 2
In the first column, we list all given numbers from the sequence in their order of appearance. Column 2, n, denotes the place where the terms appear in the sequence. Column 3, d, contains the common difference between each term, which in this case is the addition of 2 to each term, to derive the next consecutive term. Below we input the information from the tale into the equation: a + (n-1)d.
ap sequence
For the second sequence, the arithmetic progression may be expressed as:
By now, you must be wracking your brain for what possible use you might have of such a mathematical concept. Interestingly, arithmetic progression is so ubiquitous that we are constantly surrounded by it in our daily lives. On the rare but lucky occasion that your local bus is running on time, you can thank arithmetic progression. If we take the first arrival of the bus at the stop as the initial term and use every consecutive arrival at a constant interval of time to derive the common difference. Each return to the same stop is considered a value of n. Knowing the initial time, the common difference, and n, we can predict the time at which the next bus or the bus after a certain interval will arrive!
KEY TAKEAWAYS
An arithmetic progression is the sequencing of numbers in which the consecutive number is derived through a sum, and in which there is a common difference between two consecutive terms. The nth term can be derived using the formula a + (n-1)d, where a is the initial term, n is the numerical order in which the nth term appears, and d is the common difference between two consecutive terms.
In an arithmetic progression, it is possible to figure out the sum of n terms manually, using Sum of n terms in Arithmetic Progressio