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Pre-Calculus56Updated Oct 4, 20262 pages

Understanding Sequences and Series

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Sequences are patterns that follow specific rules, allowing us to predict terms and find their sums. Understanding arithmetic and geometric sequences is crucial for solving real-world problems involving patterns, growth, and accumulation.
Sequences and Series – page 1

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Sequence Formulas

Ever wonder how mathematicians predict the 100th number in a pattern without calculating all previous terms? The answer lies in sequence formulas! Arithmetic sequences increase by a constant difference (like 2, 4, 6, 8...), while geometric sequences multiply by a constant ratio (like 2, 6, 18, 54...).

For arithmetic sequences, you can find any term using the explicit formula: An=a1+(n−1)dA_n = a_1 + (n-1)d where a1a_1 is the first term, dd is the common difference, and nn is the term number. Alternatively, the recursive formula defines each term based on the previous one: a1=(first term)a_1 = \text{(first term)}, an=an−1+da_n = a_{n-1} + d.

For geometric sequences, the explicit formula is an=a1rn−1a_n = a_1r^{n-1} where rr is the common ratio. The recursive form is a1=(first term)a_1 = \text{(first term)}, an=an−1⋅ra_n = a_{n-1} \cdot r.

💡 Need to find the sum of a sequence? For arithmetic sequences, use Sn=n(a1+an2)S_n = n(\frac{a_1 + a_n}{2}). For geometric sequences, use Sn=a1(1−rn1−r)S_n = a_1(\frac{1 - r^n}{1 - r}).

When working with finite sequences, these formulas give you powerful shortcuts that eliminate the need to calculate each individual term.

Sequences and Series – page 2

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Finding Sequence Parameters

When you're given sequence terms but not told the formula, you can reverse-engineer the parameters! This skill is essential for analyzing patterns in science, finance, and other real-world applications.

For arithmetic sequences, first find the common difference dd by using an2−an1n2−n1=d\frac{a_{n_2} - a_{n_1}}{n_2 - n_1} = d where n2>n1n_2 > n_1. Once you know dd, find the first term with an1=a1+(n1−1)da_{n_1} = a_1 + (n_1 - 1)d and solve for a1a_1.

For geometric sequences, determine the common ratio rr by using ns−1an2an1^{n_s-1}\sqrt{\frac{a_{n_2}}{a_{n_1}}} where nsn_s is the number of terms between (and including) an1a_{n_1} and an2a_{n_2}. Then find the first term with an1=a1rn1−1a_{n_1} = a_1r^{n_1-1} and solve for a1a_1.

🔑 Remember that when solving sequence problems, identifying whether it's arithmetic or geometric is your crucial first step!

With these tools, you can work backwards from any terms in a sequence to discover its underlying pattern and formula.

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