Question

Show all steps please

Answer 1

D) -1001

Explanation:

`\displaystyle \sum^(22)_(k=1)(-5k+12)`

The given sigma notation means the sum of the series with nth term (-5k + 12), starting with k = 1 and ending with k = 22.

As (-5k + 12) is linear, the series is arithmetic.

The first term of the sequence is when k = 1:

`\implies a_1=-5(1)+12=7`

The last term of the sequence is when k = 22:

`\implies a_(22)=-5(22)+12=-98`

The formula for the sum of the first n terms of an arithmetic series is:

`S_n=(n)/(2)(a_1+a_n)`

Therefore, substitute a₁ = 7, a₂₂ = -98, and n = 22 into the sum formula to find the sum of the first 22 terms of the arithmetic series:

`\begin{aligned}\implies S_(22)&=(22)/(2)(7-98)\\\\&=11(-91)\\\\&=-1001 \end{aligned}`

Therefore, the sum of the first 22 terms of the arithmetic series is -1001.

Answer 2

D: -1001.

Explanation:

To find the sum of the arithmetic series, we need to add up all the terms in the series. The series is given by:

22 + (-5(1) + 12) + (-5(2) + 12) + ... + (-5(k) + 12) + ...

where k = 1 represents the first term after 22.

We can simplify the series by distributing the -5 and combining like terms:

22 + (7) + (2) + (-3) + (-8) + ... + (-5(k) + 12) + ...

Now we can express the series as a summation:

Σ (-5k + 12) = (-5(1) + 12) + (-5(2) + 12) + ... + (-5(k) + 12) + ...

We can see that this is an arithmetic series with first term a1 = 7, common difference d = -5, and k terms.

The formula for the sum of an arithmetic series is:

Sn = (n/2) * (a1 + an)

where:

n = number of terms in the series

a1 = first term

an = nth term

To find the sum of the given arithmetic series, we need to determine the value of n and an.

The value of n can be found using the formula:

an = a1 + (n - 1)d

where d = -5 and a1 = 7. We want to find the value of n such that an is the last term of the series, which is -98:

-98 = 7 + (n - 1)(-5)

-98= 7 - 5n + 5

-110 = -5n

n = 22

Since n must be a positive integer, we round up to 22.

Now we can find the value of an using the same formula:

an = a1 + (n - 1)d

an = 7 + (22 - 1)(-5)

an = -98

We can now use the formula for the sum of an arithmetic series to find the sum:

Sn = (n/2) * (a1 + an)

Sn = (22/2) * (7 - 98)

Sn = -1001

Therefore, the sum of the arithmetic series is -1001

The correct answer is option D: -1001.