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Predicting What Comes Next: Exploring Sequences - Sum of the First n Natural Numbers

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A sequence is an ordered list of numbers where each member is called a term. The sequence of natural numbers is 1,2,3,4,…1, 2, 3, 4, \dots and continues infinitely.

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Predicting the next term in a sequence involves identifying the underlying pattern or rule. In the sequence of natural numbers, the rule is to add 11 to the previous term.

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The sum of the first nn natural numbers represents the total of all integers starting from 11 up to nn. This is denoted as SnS_n.

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The formula for the sum of the first nn natural numbers is derived from the fact that if you pair the first and last terms (1+n1 + n), the second and second-to-last terms (2+(n−1)2 + (n-1)), and so on, each pair equals (n+1)(n + 1).

📐Formulae

Sn=1+2+3+⋯+n=n(n+1)2S_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}

Sn=∑i=1niS_n = \sum_{i=1}^{n} i

💡Examples

Problem 1:

Calculate the sum of the first 5050 natural numbers.

Solution:

Given n=50n = 50. We use the formula: Sn=n(n+1)2S_n = \frac{n(n+1)}{2} S50=50(50+1)2S_{50} = \frac{50(50+1)}{2} S50=50×512S_{50} = \frac{50 \times 51}{2} S50=25×51S_{50} = 25 \times 51 51×2525510201275\begin{array}{r} 51 \\ \times 25 \\ \hline 255 \\ 1020 \\ \hline 1275 \end{array} S50=1275S_{50} = 1275

Explanation:

Substitute n=50n = 50 into the sum formula to find the total of numbers from 11 to 5050.

Problem 2:

If the sum of the first nn natural numbers is 5555, find the value of nn.

Solution:

We are given Sn=55S_n = 55. Using the formula: 55=n(n+1)255 = \frac{n(n+1)}{2} 110=n(n+1)110 = n(n+1) n2+n−110=0n^2 + n - 110 = 0 Factoring the quadratic equation: (n+11)(n−10)=0(n+11)(n-10) = 0 Since nn must be a positive natural number, n=10n = 10.

Explanation:

We set up a quadratic equation by plugging the known sum into the formula and solved for the positive integer nn.

Problem 3:

Find the sum of natural numbers from 1111 to 2020.

Solution:

To find the sum from 1111 to 2020, we calculate the sum of the first 2020 numbers and subtract the sum of the first 1010 numbers: S11…20=S20−S10S_{11\dots20} = S_{20} - S_{10} S20=20(21)2=10×21=210S_{20} = \frac{20(21)}{2} = 10 \times 21 = 210 S10=10(11)2=5×11=55S_{10} = \frac{10(11)}{2} = 5 \times 11 = 55 Difference=210−55=155\text{Difference} = 210 - 55 = 155

Explanation:

The sum of a sub-sequence can be found by subtracting the sum of the unwanted leading terms from the total sum.