Bestimmen Sie für die arithmetischen Folgen das Anfangsglied a₁, die Differenz d, das allgemeine Glied aₙ und die 20. Partialsumme: a) 7, 11, 15, 19, 23, ... b) 1, 3, 5, 7, 9, ...... Bestimmen Sie für die arithmetischen Folgen das Anfangsglied a₁, die Differenz d, das allgemeine Glied aₙ und die 20. Partialsumme: a) 7, 11, 15, 19, 23, ... b) 1, 3, 5, 7, 9, ... c) 4, 4 1/2, 5, 5 1/2, 6, ... d) 20, 17, 14, 11, 8, ...
Understand the Problem
Die Frage fordert uns auf, für die gegebenen arithmetischen Folgen das Anfangsglied a₁, die Differenz d, das allgemeine Glied aₙ und die 20. Partialsumme zu bestimmen. Wir müssen diese Werte für jede der vier gegebenen Folgen (a, b, c und d) berechnen.
Answer
a) $a_1 = 7$, $d = 4$, $a_n = 4n + 3$, $S_{20} = 900$ b) $a_1 = 1$, $d = 2$, $a_n = 2n - 1$, $S_{20} = 400$ c) $a_1 = 4$, $d = 0.5$, $a_n = 0.5n + 3.5$, $S_{20} = 175$ d) $a_1 = 20$, $d = -3$, $a_n = 23 - 3n$, $S_{20} = -170$
Answer for screen readers
a) $a_1 = 7$, $d = 4$, $a_n = 4n + 3$, $S_{20} = 900$ b) $a_1 = 1$, $d = 2$, $a_n = 2n - 1$, $S_{20} = 400$ c) $a_1 = 4$, $d = 0.5$, $a_n = 0.5n + 3.5$, $S_{20} = 175$ d) $a_1 = 20$, $d = -3$, $a_n = 23 - 3n$, $S_{20} = -170$
Steps to Solve
- Determine $a_1$ for each sequence
$a_1$ is the first term in the sequence. So, we can directly read it off from the given sequences. a) $a_1 = 7$ b) $a_1 = 1$ c) $a_1 = 4$ d) $a_1 = 20$
- Determine $d$ for each sequence
$d$ is the common difference between consecutive terms. We can find it by subtracting any term from the term that follows it. a) $d = 11 - 7 = 4$ b) $d = 3 - 1 = 2$ c) $d = 4\frac{1}{2} - 4 = \frac{1}{2} = 0.5$ d) $d = 17 - 20 = -3$
- Determine $a_n$ for each sequence
The formula for the $n$th term of an arithmetic sequence is $a_n = a_1 + (n-1)d$. We can plug in the $a_1$ and $d$ that we found in the previous steps. a) $a_n = 7 + (n-1)4 = 7 + 4n - 4 = 4n + 3$ b) $a_n = 1 + (n-1)2 = 1 + 2n - 2 = 2n - 1$ c) $a_n = 4 + (n-1)0.5 = 4 + 0.5n - 0.5 = 0.5n + 3.5$ d) $a_n = 20 + (n-1)(-3) = 20 - 3n + 3 = 23 - 3n$
- Determine $S_{20}$ for each sequence
The formula for the sum of the first $n$ terms of an arithmetic sequence is $S_n = \frac{n}{2}(a_1 + a_n)$. Since we want $S_{20}$, we can use $S_{20} = \frac{20}{2}(a_1 + a_{20})$. First we need to find $a_{20}$ by plugging in $n=20$ into the $a_n$ formula a) $a_{20} = 4(20) + 3 = 80 + 3 = 83$. Thus, $S_{20} = \frac{20}{2}(7 + 83) = 10(90) = 900$ b) $a_{20} = 2(20) - 1 = 40 - 1 = 39$. Thus, $S_{20} = \frac{20}{2}(1 + 39) = 10(40) = 400$ c) $a_{20} = 0.5(20) + 3.5 = 10 + 3.5 = 13.5$. Thus, $S_{20} = \frac{20}{2}(4 + 13.5) = 10(17.5) = 175$ d) $a_{20} = 23 - 3(20) = 23 - 60 = -37$. Thus, $S_{20} = \frac{20}{2}(20 + (-37)) = 10(-17) = -170$
a) $a_1 = 7$, $d = 4$, $a_n = 4n + 3$, $S_{20} = 900$ b) $a_1 = 1$, $d = 2$, $a_n = 2n - 1$, $S_{20} = 400$ c) $a_1 = 4$, $d = 0.5$, $a_n = 0.5n + 3.5$, $S_{20} = 175$ d) $a_1 = 20$, $d = -3$, $a_n = 23 - 3n$, $S_{20} = -170$
More Information
The 20th partial sum represents the sum of the first 20 terms in the sequence. In sequence d), the terms are decreasing, so it makes sense that the 20th partial sum is negative.
Tips
- Incorrectly calculating the common difference, $d$, especially when dealing with negative differences or fractions.
- Using the wrong formula for $a_n$ or $S_n$.
- Making arithmetic errors when substituting values into the formulas.
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