Geometric sequences involve numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (denoted as r). The general formula for the n th term is an=a1⋅rn−1a_n=a_1\cdot r^{n-1}an​=a1​⋅rn−1, where a1a_1a1​ is the first term. To find the next term, divide any term by the one before it to get r , then multiply the last given term by r.

Finding the Common Ratio

  • Identify at least two consecutive terms from the sequence provided (e.g., if given 2, 6, 18, divide 6/2 = 3 or 18/6 = 3 to confirm r = 3).
  • Verify the ratio is constant across all pairs; if not, it may not be geometric.
  • Example : For 4, 12, 36, r = 12/4 = 3 , so next term is 36×3=10836\times 3=10836×3=108.

Step-by-Step Process

  1. List the given terms and compute ratios: e.g., for 5, 10, 20, ratios are 10/5=2, 20/10=2 (r=2).
  2. Multiply the final term by r : 20×2=4020\times 2=4020×2=40 (next term).
  3. Double-check with the formula: If first term a1=5 , r=2 , fourth term is 5×23=405\times 2^{3}=405×23=40.
  1. For fractions or negatives, same rules apply: e.g., 1, -2, 4 (r=-2), next is 4×−2=−84\times -2=-84×−2=−8.

Real-World Examples

Population Growth : Bacteria double every hour (sequence: 100, 200, 400...), next term 800.

Investments : $1000 at 5% compound interest yearly: 1000, 1050, 1102.50..., next ≈1157.63 (r=1.05).

Since no specific sequence was provided in your query, apply these steps to yours—share the terms for an exact calculation! This method works reliably for whole numbers, decimals, or negatives.

TL;DR : Next term = last term × common ratio (r from prior terms).

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