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Convert Finite Sequences to Summation (Sigma) Notation - Math Practice Problems
Mathematics
Grade 10 (Junior High School)
Question Content
Schreiben Sie mit Hilfe des Summenzeichens (Write using the summation symbol): (a) 1 + 3 + 9 + 27 + 81 (3 points) (b) 1 - 1 + 4/3 - 2 (3 points) (c) 1 - 1/2 + 1/4 - 1/8 (3 points)
Correct Answer
(a) $\sum_{k=0}^{4} 3^k$; (b) $\sum_{k=0}^{3} (-1)^k \cdot \frac{k+1}{1}$ (or equivalent form $\sum_{k=0}^{3} (-1)^k (k+1)$); (c) $\sum_{k=0}^{3} (-1)^k \cdot \frac{1}{2^k}$
Detailed Solution Steps
1
Step 1: Analyze part (a): Identify the pattern of the sequence. The terms are $3^0=1$, $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$. There are 5 terms, starting at $k=0$ and ending at $k=4$. So we write the sum as the summation of $3^k$ from $k=0$ to $k=4$.
2
Step 2: Analyze part (b): Identify the pattern of the sequence. The terms are $(-1)^0 \cdot 1=1$, $(-1)^1 \cdot 1=-1$, $(-1)^2 \cdot \frac{4}{3}$, $(-1)^3 \cdot 2$. We can rewrite the terms as $(-1)^k \cdot \frac{k+1}{1}$ when $k=0,1,2,3$ (check: $k=0$ gives 1, $k=1$ gives -1, $k=2$ gives $\frac{3+1}{3}=\frac{4}{3}$, $k=3$ gives $-4/2=-2$). So the sum is the summation of $(-1)^k (k+1)$ from $k=0$ to $k=3$.
3
Step 3: Analyze part (c): Identify the pattern of the sequence. The terms are $(-1)^0 \cdot \frac{1}{2^0}=1$, $(-1)^1 \cdot \frac{1}{2^1}=-\frac{1}{2}$, $(-1)^2 \cdot \frac{1}{2^2}=\frac{1}{4}$, $(-1)^3 \cdot \frac{1}{2^3}=-\frac{1}{8}$. There are 4 terms, starting at $k=0$ and ending at $k=3$. So we write the sum as the summation of $(-1)^k \cdot \frac{1}{2^k}$ from $k=0$ to $k=3$.
Knowledge Points Involved
1
Summation Notation (Sigma Notation)
Summation notation, denoted by $\sum$, is a mathematical shorthand used to write long sums of sequences concisely. It includes an index variable (usually $k$, $i$, or $n$), a lower limit (starting value of the index), an upper limit (ending value of the index), and the general term of the sequence being summed. It is used to represent finite and infinite series in algebra, calculus, and number theory.
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Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, $r$. For example, the sequence in part (a) is a geometric sequence with first term 1 and common ratio 3, and part (c) is a geometric sequence with first term 1 and common ratio $-1/2$. Geometric sequences are commonly represented and summed using sigma notation.
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Alternating Sequence
An alternating sequence is a sequence where the signs of the terms alternate between positive and negative. This is typically represented using $(-1)^k$ or $(-1)^{k+1}$ in the general term of the summation, where $k$ is the index variable. Parts (b) and (c) are alternating sequences, where the sign changes with each term.
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