Chapter 12: Q 98. (page 811)

Show that
$1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}$

Short Answer

Expert verified

It is shown that $1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}$ .

Step by step solution

Step 1. Write the given information.

The given information is:

1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}

We have to prove that $1 + 2 + . . . + (n - 1) + n$ is equal to $\frac{n (n + 1)}{2}$ .

Step 2. Prove the given equation.

Let's take:

$S = 1 + 2 + 3 + 4 + . . . + (n - 1) + n \dots (1)$

We can also write it as:

$S = n + (n - 1) + . . . + 4 + 3 + 2 + 1 \dots (2)$

Now add both the equations,

$S + S = [1 + 2 + 3 + 4 + . . . + (n - 1) + n] + [n + (n - 1) + . . . + 4 + 3 + 2 + 1] 2 S = (n + 1) + (n - 1 + 2) + . . . + (n - 1 + 2) + (n + 1) 2 S = (n + 1) + (n + 1) + . . . + (n + 1) 2 S = (n + 1) \cdot n S = \frac{(n + 1) n}{2}$

Therefore, we can say that $1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}$ .

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Show that
$1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}$

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Prove the given equation.

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Show that1+2+...+n-1+n=nn+12

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Prove the given equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Show that
$1 + 2 + . . . + (n - 1) + n = \frac{n (n + 1)}{2}$