Sum of Integers from 1 to n

In this lesson, we will learn how to find the sum of numbers from 1 to n.

We'll cover the following

What does the sum of integers from 1 to nn mean?

Natural numbers are positive numbers starting from 11. These can be written as:

1,2,3,4,5,6,7,8,9,10......1,2,3,4,5,6,7,8,9,10......

We want to write a program that takes a number and sums up all the numbers from 11 up till that number.

For example, if n=5n = 5, then the sum of numbers from 11 to 55 is: 1+2+3+4+5=151 + 2 + 3 + 4 + 5 = 15.

Mathematical Notation

The sum of all numbers up to a number is equal to the sum of that number and the sum of all the numbers before it. Let’s write it down mathematically:

∑i=15i\sum_{i=1} ^{5} i

== 55 ++ ∑i=14i\sum_{i=1} ^{4} i

== 55 ++ 44 ++ ∑i=13\sum_{i=1} ^{3}

== 55 ++ 44 ++ 33 ++ ∑i=12\sum_{i=1} ^{2}

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=5+4+3+2+1=5+4+3+2+1

Generic Mathematical Notation

∑i=1ni\sum_{i=1} ^{n} i

== nn ++ ∑i=1n−1i\sum_{i=1} ^{n-1} i

== nn ++ (n−1)(n-1) ++ ∑i=1n−2\sum_{i=1} ^{n-2}

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== nn ++ (n−1)(n-1) +(n−2)+(n-2) ...+2+1... +2+1

Implementation

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def sumTill(targetNumber) :
  # Base Case
  if targetNumber == 1 :
    return targetNumber
  # Recursive Case
  else :
    return targetNumber + sumTill(targetNumber - 1)
# Driver Code
targetNumber = 5
print(sumTill(targetNumber))

Explanation

Let’s begin by breaking down the solution into subtasks:

sumofnumbersfrom1to5=5+sumofnumbersfrom1to4sum \: of \: numbers \: from \: 1 \: to \: 5 = 5 + sum \: of \: numbers \: from \: 1 \: to \: 4 sumofnumbersfrom1to4=4+sumofnumbersfrom1to3sum \: of \: numbers \: from \: 1 \: to \: 4 = 4 + sum \: of \: numbers \: from \: 1 \: to \: 3

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sumofnumbersfrom1to1=1sum \: of \: numbers \: from \: 1 \: to \: 1 = 1

The last statement where execution halts, will now become our base case and the rest will be mapped to recursive case: sumofnumbersfrom1ton=n+sumofnumbersfrom1ton−1sum \: of \: numbers \: from \: 1 \: to \: n = n + sum \: of \: numbers \: from \: 1 \: to \: n - 1

Let’s have a look at an another example in the next lesson.