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neurologyYou need to know

• A linear (arithmetic) sequence is a list of numbers where the difference between consecutive terms is constant.
• The constant difference in a linear sequence is called the common difference `d`.
• An increasing linear sequence has a positive common difference `d > 0`, while a decreasing linear sequence has a negative common difference `d < 0`.
• Term numbers are indexed from `n = 1`, so `a_1` denotes the first term, `a_2` the second, and so on.
• The term-to-term rule for a linear sequence can be written as `a_{n+1} = a_n + d`.
• A position-to-term (nth term) rule for a linear sequence can be written in the form `a_n = d n + c`, where `d` is the common difference and `c` is a constant.
• For a linear sequence with nth term `a_n = d n + c`, the first term is `a_1 = d + c`.
• For a linear increasing sequence, the nth term can be found by multiplying the common difference by `n` and adding the value needed to produce the first term (i.e., `c = a_1 - d`).
• If the first term `a_1` and common difference `d` are known, then the nth term is `a_n = a_1 + (n - 1)d`.
• A sequence is linear if and only if the first differences between consecutive terms are all equal.
• The term-to-term rule can be expressed in words (e.g., “add three each time” or “subtract 0.5 each time”) and in notation (e.g., `a_{n+1} = a_n + 3` or `a_{n+1} = a_n - 0.5`).
• Linear sequences with fractions (e.g., `1/2, 3/2, 5/2, ...`) and with decimals (e.g., `0.2, 0.7, 1.2, ...`) have the same constant-difference structure as integer sequences.
• To continue a linear sequence, each subsequent term is found by adding the common difference to the previous term, which means subtracting the magnitude of the common difference if the sequence is decreasing.
• Missing terms in a linear sequence can be found by applying the common difference forwards or backwards the required number of steps.
• Given an nth-term rule `a_n = d n + c`, any specific term is found by substituting the chosen value of `n` and evaluating.
• The common difference `d` equals the difference between any pair of consecutive terms, for example `d = a_{n+1} - a_n`.
• When comparing a proposed nth-term rule with a sequence, substituting `n = 1, 2, 3` must reproduce the given first terms exactly to confirm correctness.
• In a decreasing linear sequence, the nth term still has the form `a_n = d n + c`, but `d` is negative so terms get smaller as `n` increases.

• Describe a linear pattern fluently in words, stating both the starting value and the constant step (e.g., “start at 4 and add 1.5 each time”).
• Continue a linear sequence by generating the next three terms accurately using the common difference, including with fractions and decimals.
• Find missing terms within a linear sequence by stepping forwards or backwards the correct number of intervals using the common difference.
• Determine the common difference by subtracting consecutive terms and justify linearity by showing the difference is constant.
• State a term-to-term rule in both forms: verbal (e.g., “subtract 0.25 each time”) and symbolic `a_{n+1} = a_n + d`.
• Derive the nth term of a linear increasing sequence by calculating `d`, then finding `c = a_1 - d`, and writing `a_n = d n + c`; verify by checking the first three terms.
• Generate the first five terms from a given nth-term rule by substituting `n = 1, 2, 3, 4, 5` and evaluating correctly.
• Find any specified term (e.g., the 20th term) by substituting the given `n` into the nth-term expression and computing the value accurately.