Answer

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Hint: Gauss method is applicable only to arithmetic progression and from this we can conclude that the sequence is in AP and apply the suitable formula to find the sum of the given series.

Complete step-by-step answer:

The given series is -5,-4,-3,-2,-1,0,1,2………

We can conclude and say that this series is in Arithmetic Progression

Since, ${T_3} - {T_2} = {T_2} - {T_1}$ = -4-(-5)=1=d=common difference

We have to find out the sum of first 100 terms of the series using Gauss method

We know that Gauss formula is given by ${S_n} = \dfrac{n}{2}[first{\text{ }}term + last{\text{ }}term]$

In the given series the first term=-5, last term is unknown

So, let us find out the nth term (last term) by making use of ${T_n}$ formula of AP

We know that the nth term ${T_n}$ of an AP is given by ${T_n} = a + (n - 1)d$

Here a=-5, n=100, d=1

Let’s substitute these values in the formula

So, we get ${T_{100}}$ =-5+(100-1)1

=-5+99

${T_{100}}$ =94=last term

We have to find out the sum of the first 100 terms by Gauss formula

${S_n} = \dfrac{n}{2}[first{\text{ }}term + last{\text{ }}term]$

Here a=-5, d=1, last term=100

So, we can write

${S_{100}} = \dfrac{{100}}{2}[ - 5 + 94]$

=50[89]

=4,450

So, we can write ${S_{100}} = 4450$

Note: In this question ,we have been asked to find out the sum by Gaussian method , if it was not mentioned, we can find out the sum of the series by making use of an alternate formula.

Complete step-by-step answer:

The given series is -5,-4,-3,-2,-1,0,1,2………

We can conclude and say that this series is in Arithmetic Progression

Since, ${T_3} - {T_2} = {T_2} - {T_1}$ = -4-(-5)=1=d=common difference

We have to find out the sum of first 100 terms of the series using Gauss method

We know that Gauss formula is given by ${S_n} = \dfrac{n}{2}[first{\text{ }}term + last{\text{ }}term]$

In the given series the first term=-5, last term is unknown

So, let us find out the nth term (last term) by making use of ${T_n}$ formula of AP

We know that the nth term ${T_n}$ of an AP is given by ${T_n} = a + (n - 1)d$

Here a=-5, n=100, d=1

Let’s substitute these values in the formula

So, we get ${T_{100}}$ =-5+(100-1)1

=-5+99

${T_{100}}$ =94=last term

We have to find out the sum of the first 100 terms by Gauss formula

${S_n} = \dfrac{n}{2}[first{\text{ }}term + last{\text{ }}term]$

Here a=-5, d=1, last term=100

So, we can write

${S_{100}} = \dfrac{{100}}{2}[ - 5 + 94]$

=50[89]

=4,450

So, we can write ${S_{100}} = 4450$

Note: In this question ,we have been asked to find out the sum by Gaussian method , if it was not mentioned, we can find out the sum of the series by making use of an alternate formula.

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