Question

Find the sum:

(i) 2 + 4 + 6 ... + 200

(ii) 3 + 11 + 19 + ... + 803

(iii) (−5) + (−8)+ (−11) + ... + (−230)

(iv) 1 + 3 + 5 + 7 + ... + 199

(v) $7+10\frac{1}{2}+14+...+84$

(vi) 34 + 32 + 30 + ... + 10

(vii) 25 + 28 + 31 + ... + 100

(viii) $18+15\frac{1}{2}+13+...+\left(-49\frac{1}{2}\right)$

Answer 1

In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum of n terms of an A.P.,

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

(i)

Common difference of the A.P. (d) =

So here,

First term (a) = 2

Last term (l) = 200

Common difference (d) = 2

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Further simplifying,

Now, using the formula for the sum of n terms, we get

On further simplification, we get,

Therefore, the sum of the A.P is

(ii)

Common difference of the A.P. (d) =

So here,

First term (a) = 3

Last term (l) = 803

Common difference (d) = 8

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Further simplifying,

Now, using the formula for the sum of n terms, we get

Therefore, the sum of the A.P is

(iii)

Common difference of the A.P. (d) =

So here,

First term (a) = −5

Last term (l) = −230

Common difference (d) = −3

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Now, using the formula for the sum of n terms, we get

Therefore, the sum of the A.P is

(iv)

Common difference of the A.P. (d) =

So here,

First term (a) = 1

Last term (l) = 199

Common difference (d) = 2

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Now, using the formula for the sum of n terms, we get

On further simplification, we get,

Therefore, the sum of the A.P is

(v)

Common difference of the A.P is

(d) =

So here,

First term (a) = 7

Last term (l) = 84

Common difference (d) =

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Further solving for n,

Now, using the formula for the sum of n terms, we get

On further simplification, we get,

Therefore, the sum of the A.P is

(vi)

Common difference of the A.P. (d) =

So here,

First term (a) = 34

Last term (l) = 10

Common difference (d) = −2

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Further solving for n,

Now, using the formula for the sum of n terms, we get

On further simplification, we get,

Therefore, the sum of the A.P is

(vii)

Common difference of the A.P. (d) =

So here,

First term (a) = 25

Last term (l) = 100

Common difference (d) = 3

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

Further solving for n,

Now, using the formula for the sum of n terms, we get

On further simplification, we get,

Therefore, the sum of the A.P is.
(viii) $18+15\frac{1}{2}+13+...+\left(-49\frac{1}{2}\right)$

Common difference of the A.P. (d) =

= $$\begin{gathered} =15\frac{1}{2}-18 \\ =\frac{31}{2}-18 \\ =\frac{31-36}{2} \\ =\frac{-5}{2} \end{gathered}$$

So here,

First term (a) = 18

Last term (l) = $-49\frac{1}{2}=\frac{-99}{2}$

Common difference (d) = $\frac{-5}{2}$

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

So, for the last term,

$$\begin{gathered} \frac{-99}{2}=18+\left(n-1\right)\frac{-5}{2} \\ \frac{-99}{2}=18+\left(\frac{-5}{2}\right)n+\frac{5}{2} \\ \frac{5}{2}n=18+\frac{5}{2}+\frac{99}{2} \\ \frac{5}{2}n=18+\frac{104}{2} \\ n=28 \end{gathered}$$

Now, using the formula for the sum of n terms, we get

$$\begin{gathered} S_n=\frac{28}{2}\left[2\times 18+\left(28-1\right)\left(\frac{-5}{2}\right)\right] \\ {S}_n=14\left[36+27\left(\frac{-5}{2}\right)\right] \\ {S}_n=-441 \end{gathered}$$

Therefore, the sum of the A.P is $S_n=-441$.