Question

Determine the sum : 1 + (-2) + 3 + (-4) + 5 (-6)+....... +(-50)

Give the correct answer ​

Answer 1

Answer:

(- 25)

Step-by-step explanation:

Given :- 1 + (-2) + 3 + (-4) ......... + (-50)

To determine the sum of given series.

Here, if we break it in two A.P., it would be easier for us to solve.

1 + (-2) + 3 + (-4)..... + (-50), can be written as :-

1 + 3 + 5...... + 49 + (-2) + (-4) ....... + (-50)

Now, the sum of given series would be equal to the sum of these two A.P.

for, first A.P. we have

a = 1

d = 3 - 1 = 2

last term = 49

We will need to find the number of terms to find the sum.

So, $\sf T_n = a + (n - 1)d$

→ 49 = 1 + (n - 1)2

→ 49 = 1 + 2n - 2

→ 49 = 2n - 1

→ 49 + 1 = 2n

→ n = 50/2

→ n = 25 terms.

So, sum = $\sf \frac{n}{2}(a + l)$

where l is the last term

So, sum = 25/2 × (1 + 49)

→ 25/2 × 50

→ 25 × 25

= 625

Now, for second A.P.

a = -2

d = -4 - (-2) = -4 + 2 = -2

last term = - 50

So, again, number of terms =

-50 = -2 + (n - 1)(-2)

→ - 50 = -2 - 2n + 2

→ - 50 = - 2n

→ n = 25

So, sum = 25/2 × (-2 - 50)

→ sum = 25/2 × -52

→ sum = 25 × -26

→ sum = -650

Now, total sum = 625 + (-650) = -25

Hence, the answer is -25.

$\rule{200}2$

Answer 2

Solution:-

Let 'x' be the sum of Given Series.

=) x = 1 + (-2) + 3 + (-4) + 5 + (-6)+....... +(-50)

=) x = 1 + (-2) + 3 + (-4) + 5 + (-6)+....... + 49 +(-50)

Let's Break it into two A.P. Separating the positive term from Negative terms. we get,

=) x = [ 1 + 3 + 5 +................ + 49 ] + [ -2 -4 -6 -.................. - 50 ]

=) x = [ 1 + 3 + 5 +................ + 49 ] - [ + 2 + 4 + 6 + .................. + 50 ]

Let's Assume that,

y = [ 1 + 3 + 5 +................ + 49 ]

and z = [ 2 + 4 + 6 + .................. + 50 ]

So,

[x = y - z]________________(1)

Total Number of Terms = 50.

The Alternative values are in the two A.Ps , So Number of term in one A.P = 50/2 = 25

For y,

a ( First Term) = 1

d ( Common Difference) = 2

l ( Last term) = 49

By using the Formula,

Sn = n/2 ( a + l )

=) y = 25/2 ( 1 + 49)

=) y = 25/2 × 50

=) y = 25 × 25

=) y = 625

For z,

a = 2

d = 2

l = 50

Now,

z = n/2 ( a + l)

=) z = 25/2 ( 2 + 50)

=) z = 25/2 × 52

=) z = 25 × 26

=) z = 650

Substituting the value of y and z in eq(1). we get,

=) x = y - z

=) x = 625 - 650

=) x = -25

Hence,

The Sum of the Given Series is -25.