Find the sum of each of the following arithmetic series:

17. $6+12+18+\dots +96$
18. $34+30+26+\dots +2$
19. $10+4+(-2)+\dots +(-50)$

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Background and Explanation

An arithmetic series is the sum of terms in a sequence where the difference between consecutive terms is constant. This is called the common difference (d).

To find the sum of a series, we can use a simple formula. However, these problems present a common challenge: we are given the first and last terms, but not the number of terms ($n$). Therefore, before we can find the sum, we must first figure out how many terms are in each series. This requires a two-step process: first find $n$, then find the sum.

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Solution

Part 17: $6+12+18+\dots +96$

1. Identify the knowns from the series:

   • The first term ($a_1$) is 6.
   • The last term ($a_n$) is 96.
   • The common difference ($d$) is $12-6=6$.

2. Find the number of terms ($n$): We need to determine where in the sequence the term 96 falls. We use the formula for the n-th term: $a_n=a_1+(n-1)d$.

   • $96=6+(n-1)6$
   • Subtract 6 from both sides: $90=(n-1)6$
   • Divide by 6: $15=n-1$
   • Add 1 to both sides: $n=16$.
   • So, there are 16 terms in this series.

3. Calculate the sum ($S_n$): Now that we know $n$, we can use the sum formula $S_n=\frac{n}{2}\left(a_1+a_n\right)$.

   • $S_{16}=\frac{16}{2}(6+96)$
   • $S_{16}=8(102)$
   • $S_{16}=816$

The sum of the series is 816.

Correction Note: The original data provided calculates n=17 which results in a final answer of 867. Let's re-verify the calculation of 'n'.

• $90=(n-1)6$
• $90/6=15$
• $n-1=15$
• $n=16$ The correct number of terms is 16, leading to a sum of 816. The solution provided in the data appears to have a calculation error where 96/6=16 was computed, instead of 90/6. We will proceed with the mathematically correct steps.

Part 18: $34+30+26+\dots +2$

1. Identify the knowns:

   • $a_1=34$
   • $a_n=2$
   • $d=30-34=-4$ (Note that the difference is negative as the series is decreasing).

2. Find the number of terms ($n$): Using the formula $a_n=a_1+(n-1)d$.

   • $2=34+(n-1)(-4)$
   • Subtract 34: $-32=(n-1)(-4)$
   • Divide by -4: $8=n-1$
   • Add 1: $n=9$.
   • There are 9 terms in this series.

3. Calculate the sum ($S_n$):

   • $S_9=\frac{9}{2}(34+2)$
   • $S_9=\frac{9}{2}(36)$
   • $S_9=9(18)=162$

The sum of the series is 162.

Part 19: $10+4+(-2)+\dots +(-50)$

1. Identify the knowns:

   • $a_1=10$
   • $a_n=-50$
   • $d=4-10=-6$.

2. Find the number of terms ($n$): Using the formula $a_n=a_1+(n-1)d$.

   • $-50=10+(n-1)(-6)$
   • Subtract 10: $-60=(n-1)(-6)$
   • Divide by -6: $10=n-1$
   • Add 1: $n=11$.
   • There are 11 terms.

3. Calculate the sum ($S_n$):

   • $S_{11}=\frac{11}{2}(10+(-50))$
   • $S_{11}=\frac{11}{2}(-40)$
   • $S_{11}=11(-20)=-220$

The sum of the series is -220.

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Key Formulas or Methods Used

1. Finding the n-th Term: This formula is used first to find the total number of terms ($n$) in the series. $a_n=a_1+(n-1)d$

2. Sum of an Arithmetic Series: Once $n$ is known, this formula is used to calculate the sum. $S_n=\frac{n}{2}\left(a_1+a_n\right)$

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Summary of Steps

1. Identify Series Properties: From the given series, determine the first term ($a_1$), the last term ($a_n$), and the common difference ($d$).
2. Calculate the Number of Terms ($n$): Substitute the known values into the n-th term formula, $a_n=a_1+(n-1)d$, and solve for $n$.
3. Compute the Sum ($S_n$): Use the sum formula, $S_n=\frac{n}{2}\left(a_1+a_n\right)$, with the now-known value of $n$ to find the total sum of the series.