# Read finite geometric sums in Book Isle: separate the fifth term from the five-term total
Language: en
Canonical: [Read finite geometric sums in Book Isle: separate the fifth term from the five-term total]()
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Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T09:48:08+08:00
Author: Bison
Derive a five-term sum by shifted subtraction, then check ratio one and finite halving without replacing them by infinite sums.
English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.
A last term of forty-eight does not make the five-term sum forty-eight. Read original geometric cards in Book Isle, state count and ratio, then derive the total by shifted subtraction. Check the finite formula, its exception and finite/infinite boundaries. External notes suffice without automatic series calculation or completion of ellipses.
## List five terms and fix the first
Use first term three, ratio two and five terms: three, six, twelve, twenty-four, forty-eight. Number from one; the last exponent is four because the first term has not yet been doubled. Record source and scope rather than infer count from an ellipsis. Read text or PDF and calculate externally. A zero-indexed source needs an explicit mapping instead of silently adopting this numbering.
## Add directly as a control
Calculate successive totals nine, twenty-one, forty-five and ninety-three. Separate last term forty-eight from total ninety-three. Five terms permit a direct independent check, rather than copying a formula result and calling two methods consistent. Return to unclear operators instead of filling them from a familiar exercise. Count, values and operations all belong in the record.
## Write shifted rows and cancel corresponding terms
Name the sum S. Doubling yields six plus twelve plus twenty-four plus forty-eight plus ninety-six. Subtract the original row: four equal terms cancel, leaving two S minus S equals ninety-six minus three, so S is ninety-three. Align by values and multiplication, not display wrapping. Cancellation removes matching terms, not arbitrary neighbours. Ninety-six is outside the original five terms but has a defined derivational role.
## State the general formula and condition
For first term a, ratio r and n terms use `S=a*(r^n-1)/(r-1)`, with r not one. The exponent n comes from the extra shifted term; the nth original term has exponent n minus one. Substitution three/two/five gives three times thirty-one over one, ninety-three. Retain variables, boundaries and brackets rather than memorise only the denominator. A displayed formula is not app validation of algebra.
## Handle ratio one without division by zero
A control with first term three, ratio one and five terms totals fifteen. Do not substitute into the previous fraction and turn zero-over-zero into zero, or declare ratio-one geometric sequences nonexistent. That expression is inapplicable directly. Verify count times first term and attach the exception to the formula. This card no longer has last term forty-eight; use a new identifier so five positions do not invite stale values. Conditions determine permitted operations.
## Keep a halving card finite
For first term three, ratio one half and five terms, use three, one point five, point seven five, point three seven five and point one eight seven five. Their sum is ninety-three sixteenths, five point eight one two five. Infinite continuation can sum to six, but this card has five terms only. Its deficit is three sixteenths. Keep the original doubling and new halving cards distinct, with count and ratio attached to each result.
## Check exponents and infinite-sum substitution
A total one hundred eighty-nine suggests a sixth term ninety-six was added. A last term ninety-six calls for checking index origin. Replacing the finite halving total with six ignores count; closeness is not exact equality. After wrapping, verify the whole numerator and denominator. Share original terms, shifted subtraction and conditions rather than a whole workbook. Record formatting and arithmetic errors separately without claiming automatic correction or sum-type recognition.
## Reopen and check all three cards
Reopen and identify doubling, constant-three and halving cards. Recalculate ninety-three, fifteen and ninety-three sixteenths without answers. Retain last-term/total distinction, derivation, r-not-one condition and finite/infinite boundary. Use bookmarks or example identifiers after font changes, not display line count as n. Recalculate changed count or first term; a sequence name does not fix one permanent sum. Deliver reproducible reasoning rather than only a calculator decimal.
## Related guides and sources
[Read sequences in Book Isle: separate differences, ratios and term indices](); [Book Isle: Match summation indices and limits to expanded terms]().
Checked on November 15, 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 controls in the installed version.
Method reference: [OpenStax: finite geometric series](). Materials and numbers are original exercises. Record external reports, paper reasoning and actual reading observations separately, without inventing missing conditions. Sources do not establish new automatic analysis by the app. No step-by-step physical-device, private-server or all-output-environment testing is claimed.