What this calculator does
A geometric series is a sum of terms in which each term is found by multiplying the previous one by a fixed number called the common ratio. The sequence looks like a, ar, ar², ar³, …, ar^(n−1). This calculator finds two things at once: the nth term of the progression and the sum of its first n terms (the partial sum), given the first term a, the common ratio r, and the number of terms n.
How to use it
Enter the first term a (any real number, positive, negative, or fractional), the common ratio r, and the number of terms n (a positive whole number). Optionally choose a display precision to control how many significant digits are shown — this affects the display only, not the calculation. Press calculate to see both the nth term aₙ and the sum Sₙ.
The formula explained
The nth term is a_n = a · r^{ n-1}. For the sum, when r is not equal to 1 we use the closed form
Worked example
With a = 1, r = 2, n = 10: the 10th term is a_n = 1 · 2^{9} = 512. The sum is
使用注意事項與精準校對原則（6 大核心守則）
📐 1. 嚴格對齊輸入量綱與單位
輸入各項數值時請務必確認量綱（例如小時、天數、金額、百分比），避免單位不匹配引發推算偏差。
🔒 2. 純前端本機毫秒級隱私保障
所有計算邏輯 100% 在您的終端瀏覽器內完成，資料絕不上傳伺服器，嚴格守護商業與個人隱私。
📊 3. 符合公認學術與產業計算模型
演算法嚴格遵循國際通用數學公式與標準計算模型，經多重情境交叉核對，數值嚴密精確。
📋 4. 一鍵複製與免登入 PDF 匯出
支援一鍵複製推算結果字串至剪貼簿，亦可直接點擊列印按鈕輸出純淨 PDF 工作報表。
🔄 5. 多情景敏感度動態對比
建議針對極端值與多組預期情景分別試算，觀察輸出數據的波動區間，輔助做出更穩健的決策。
🛡️ 6. 專業諮詢與日常參考免責聲明
本工具運算結果供日常排程、學業研習與事前參考之用，如涉及重大法規或投資決策請諮詢持證專家。