Question

Calculate 1-2+3-4+5-6+7-8+..........+19-20

Answer 1

Answer:

$- 1 - 1 - 1........10 \: times = - 10$

Answer 2

$\bf 1 - 2 + 3 - 4 + ... + 19 - 20 = - 10$

Given:

• $1 - 2 + 3 - 4 + 5 - 6 +... + 19 - 20$

To find:

• Calculate the sum.

Solution:

Formula/concept to be used:

Sum of n terms of AP:

$\bf S_n = \frac{n}{2} (a + l)$

General term of AP:

$\bf a_n = a + (n - 1)d$

Step 1:

Divide the series into two parts.

$1 - 2 + 3 - 4 + 5 - 6 + ... + 19 - 20 \\ = (1 + 3 + 5 + 7 + ... + 19) \\ - (2 + 4 + 6 + 8 + ... + 20)$

Step 2:

Calculate the sum of the first series.

$1 + 3 + 5 + 7 + ... + 19$

Find the number of terms in the series.

In this AP;

a= 1

d= 2

$\bf a_n = 19$

$19 = 1 + (n - 1)2$

$2(n - 1) = 18$

$n - 1 = 9$

$\bf n = 10$

Sum of terms of first AP

$S_1 = \frac{10}{2} (1 + 19)$

$S_1 = 5 \times 20$

$\bf S_1= 100$

Step 3:

Calculate the sum of the second series.

$2 + 4 + 6 + 8 + ... + 20$

here

$\bf a = 2 \\ \bf d=2$

$\bf a_n = 20$

$20 = 2 + (n - 1)2$

$2(n - 1) = 18$

$\bf n = 10$

Calculate the sum of terms of this AP:

$S_2 = \frac{10}{2} (2 + 20)$

$S_2 = 5 \times 22$

$\bf S_2 = 110$

Step 4:

Find the sum of the given series.

$\bf S= S_1-S_2$

$S= 100 - 110$

$\bf S= - 10$

Thus,

$\bf 1 - 2 + 3 - 4 + ... + 19 - 20 = - 10$

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