Input Data :
Number of Terms (`n`) = 100
First Term (`a`) = 1
Difference (`d`) = 1
Objective :
Find `n^{th}` term and `n^{th}` partial sum of an arithmetic sequence?
Formula :
`n^{th}` Term (`T_n`) `= a + (n - 1)d`
`n^{th}` Partial Sum (`S_n`) `= n/2(a + T_n)`
Solution :
`n^{th}` Term (`T_n`) = 1 + (100- 1)1
= 1 + (99) x 1
= 1 + 99
`T_n` = 100
`n^{th}` Partial Sum (`S_n`)` = 100/2 (1 + 100)`
` = 100/2 \times101`
` = 10100/2`
`S_n` = 5050
Arithmetic progression calculator calculates the `n^(th)` term and the `n^(th)` partial sum of an arithmetic progression.
It is necessary to follow the next steps:
A progression is a function with positive integers as its domain. The terms of a progression belong to the range of the function. The first term of a progression is usually denoted by `a_1`, the second term by `a_2`, and so on up, the nth term by an. A progression `(a_(n))_(n∈N)` such that each term is obtained from the preceding one by adding a constant is an arithmetic progression or an arithmetic sequence. In other words, any arithmetic progression has the form
In the problems which require to find the next term in an arithmetic progression, first we find the common difference by subtracting any term from its succeeding term, then we add the common difference to the last term. For instance, the `n^(th)` term `a_n` is equal to $a_{n-1}+d$, where `a_(n−1)` is the `(n − 1)^(th)` term. Therefore,
Sum of First n Numbers | |
---|---|
Sum of first 100 natural numbers | 5050 |
Sum of first 50 even numbers | 2550 |
Sum of first 50 odd numbers | 2500 |
Sum of first 50 natural numbers | 1275 |
Sum of first 100 odd numbers | 10000 |
Sum of first 100 even numbers | 10100 |
Sum of first 10 numbers | 55 |
Sum of first 10 odd numbers | 100 |
Sum of first 10 even numbers | 110 |
Sum of first 25 natural numbers | 325 |
Sum of natual numbers from 51 to 100 | 3775 |
Sum of natual numbers from 50 to 100 | 3825 |
Sum of natual numbers from 20 to 50 | 1085 |
Arithmetic progressions are very important in mathematics. They have many useful applications in physics, statistics, finance, etc.
In working with linear functions, when we increase the `x`-value by `1` unit, the `y`-value increases by a fixed number, equal to the slope. The next term in an arithmetic progression is derived by adding the common difference to the last term. So, arithmetic progressions are something like linear functions, with the common difference as the slope.
The partial sum of an arithmetic progression has an application in the theory of numbers. For instance, the triangular numbers are the sums of the consecutive positive integers.The first triangular number is `1`, the second is the sum of `1+2=3`, the third is `1+2+3=6`, the fourth is `1+2+3+4=10` and so on. The sequence of the triangular numbers is the sequence of the partial sums of the arithmetic sequence `1,2,3,4\ldots`
Practice Problem 1:
We have `\$1000` and go to the bank to deposit money. The bank gives us the following option:
the first month we receive `\$1000`, the second month we receive `\$1080`, the third month we receive `\$1160`, etc. How much money will we receive after `15` months?
Practice Problem 2:
Given the sequence by the recurrence relation `a_{n+1} = a_n − 5` and `a_1 = 10`. Find the sum of the first `15` terms of the arithmetic sequence.
The arithmetic progression calculator, formulas for the `n^{th}` term of the sequence and the sum of `n` numbers of the sequence, step by step calculation and practice problems would be very useful for grade school students (K-12 education) in studying series and sequences and in solving real world problems.