# Read sequences in Book Isle: separate differences, ratios and term indices Language: en Canonical: [Read sequences in Book Isle: separate differences, ratios and term indices]() 中文：[中文 HTML]() English: [English HTML]() Published: 2026-11-12T00:00:00+08:00 Modified: 2026-11-12T09:43:29+08:00 Author: Bison Work through two original sequences starting at three, checking first terms, negative ratios and the limits of inferring an infinite rule from a few values. English translation of the [Chinese original](), translated on 2026-11-12. Original publication dates, attribution, code and version notes are retained. A sequence beginning three, six, twelve is not arithmetic merely because every term grows. Read a sequence with an explicit definition in Book Isle, separating term index, value and neighbouring relations. Write the reasoning in personal notes. The app provides reading and return access; it is not asked to recognise sequences automatically, and a finite guess is not an infinite proof. ## Copy the first term and index origin Record the book, edition, chapter and example location, then check whether indexing begins at zero or one. Here a1 is explicitly three and each later term is twice its predecessor. In the control, b1 is also three but every later term adds three. Index one does not mean value one, and a subscript is not multiplication. Copy each rule separately without consulting the answers. If only three terms are listed and no continuation is defined, record a candidate pattern rather than adding an unwritten forever condition. ## Generate five terms at matching indices Starting from their own first terms, a gives three, six, twelve, twenty-four and forty-eight; b gives three, six, nine, twelve and fifteen. Matching first two terms do not make the sequences identical: the third distinguishes the rules. For each step record the predecessor and operation. Do not put a fourth value of twenty-four after b’s third term. If you cross columns, restart from the last confirmed term instead of filling gaps backwards from an answer. These are abstract numbers, not investment, growth or achievement predictions. ## Check differences instead of mere increase For b, successor minus predecessor gives three, three, three and three, matching the stated arithmetic rule. For a the differences are three, six, twelve and twenty-four, so they are not constant. Keep subtraction order fixed. Using six minus three and then six minus twelve would manufacture a sign change. Attach each difference to its two indices and return to the text when one differs. An upward or increasingly steep plotted line cannot replace pairwise checks. ## Check denominators before taking ratios Every successor-to-predecessor ratio in a is two, consistent with the common ratio. For b the first three ratios are two, one and a half, and four thirds, which are not constant. To avoid mishearing fractions, write six divided by three, nine divided by six and twelve divided by nine before evaluating them. If the predecessor is zero, do not divide by zero to infer a ratio; check the supplied multiplication recurrence separately. A zero difference and a zero ratio differ, and undefined does not mean zero. ## Relate multiplication count to index minus one With the first term indexed one, a’s nth term is three times two raised to (n−1), written `a_n=3*2^(n-1)`. Moving from term one to two multiplies once; reaching term five multiplies four times. At n equal to one the exponent is zero and the value is three; at five it is forty-eight. Using exponent n makes the first value six and shifts the sequence. For b use three plus three times (n−1), written `b_n=3+3*(n-1)`, giving fifteen at term five. Check the first term and a later term, rather than trusting one convenient substitution. If a separate convention gives value three at index zero, the exponent becomes n. Its index-one value is six rather than the old convention’s three. Map the conventions explicitly instead of mixing their indices. ## A negative ratio need not mean steady decrease For a further control let c begin at two with ratio negative two. Its first five terms are two, negative four, eight, negative sixteen and thirty-two. Signs alternate while magnitudes grow; this is neither subtracting two each time nor becoming steadily smaller. Put multiplication by negative two beside subtraction of two: subtraction instead produces zero, negative two and negative four after the initial two. The minus is part of the definition, not a connecting dash. If enlargement cannot clarify it, consult a clearer original instead of guessing. This checks the operation and does not label every geometric sequence as growth. ## Check ellipses and finite evidence If the book merely lists three, six, twelve and an ellipsis, many rules can fit those terms; appearance alone does not force twenty-four next. Look for an explicit geometric assumption, recurrence, or instruction to conjecture, distinguishing definition from guess. If a formula and recurrence disagree, keep the first conflicting index and both values, then check the starting index and brackets without editing the source. Numeric searches may find another example. Verify its problem number and context before treating a similar triple as part of the current definition. ## Leave and return to reconstruct from rules Leave reading and return to the example. Cover your results, rebuild five terms of a and b by index, and explain why the first exponent is zero. Check c’s signs and one division-by-zero restriction. Retain location, first term, index convention, recurrence, difference or ratio checks and unspecified conditions. Success means reconstructing from the definition, not memorising forty-eight or naming every list. If the next page starts at index zero, create a new mapping instead of reusing n minus one. Keep the notes identifiable as your reasoning, not automatic Book Isle calculations. ## Related guides and sources [Book Isle: Match summation indices and limits to expanded terms](); [Read function composition with Book Isle: pass the inner output to the outer function](). Checked on November 12, 2026: [Apple public information]() lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the [official App Store]() for published capabilities; verify specific controls in the installed version. Method reference: [OpenStax: recursive and explicit forms of geometric sequences](). Samples, passages and controls are original exercises. Sources establish definitions and checking methods rather than new automatic analysis in the app. Manual records may be kept externally, and unknown fields should remain explicit. This article does not claim step-by-step testing on a physical device, private server or every output environment.