How this calculation works
A geometric sequence multiplies by the same ratio at every step. If the ratio is 1, the terms are constant and the sum equals the first term multiplied by the count.
Nth term = first × ratio^(n − 1); sum = first × (1 − ratio^n) ÷ (1 − ratio)A worked example
For 2, 6, 18, 54, 162, the fifth term is 162 and the sum is 242.
Good to know
Uses the stated formula and assumptions. Displayed decimal values are rounded; use consistent units for all inputs.
A fixed multiplier creates a very different pattern
Starting at 2 and multiplying by 3 gives 2, 6, 18, 54 and 162. The differences grow even though the ratio stays fixed. That is why multiplying sequences can outgrow an additive sequence quickly.
A ratio between zero and one shrinks the magnitude instead. With ratio 1/2, terms halve repeatedly. For an infinite sequence with absolute ratio less than one, a finite limiting sum can exist, but this tool calculates only the specified finite number of terms.
Special ratios deserve a closer look
When the ratio is 1, every term is identical and the usual fraction for the sum would divide by zero. The tool handles that case separately as first term × count. A negative ratio alternates the signs of successive nonzero terms.
Large ratios or long sequences can overflow numerical limits. Near a ratio of 1, subtracting nearly equal floating-point quantities can also reduce precision, so independently verify sensitive calculations.
A few questions, answered
Does this calculate an infinite sum?
No. It returns the nth term and the sum of the first n terms only.
Why do some terms change sign?
A negative common ratio flips the sign at each multiplication.