Percentage change
Find how much a positive starting value increased or decreased. Use the same measurement unit for both entries.
MAKE THE FORMULA CLEAR
Arithmetic and geometric sequences
Find a final term and sum for an arithmetic or geometric sequence. Choose whether you repeatedly add a difference or multiply by a ratio.
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nth term
19
The first term is item 1. The nth term and sum of the first n terms are different answers; read the result labels.
Sum of the first n terms100
Number of terms10
Arithmetic: aₙ = a₁ + (n−1)d and Sₙ = n(a₁+aₙ)/2. Geometric: aₙ = a₁r^(n−1) and Sₙ = a₁(1−rⁿ)/(1−r); when r = 1, Sₙ = na₁. Adds terms individually to avoid subtraction errors near r = 1. Negative and zero common ratios are allowed; results with absolute value above 10^18 are refused.
Method and scope
Arithmetic term n is a1 + (n-1)*d and its sum is n*(a1+an)/2. Geometric term n is a1*r^(n-1); the sum is a1*(1-r^n)/(1-r), or n*a1 when r is one.
A difference is an amount added each time; a ratio is a multiplier. Zero and negative geometric ratios are allowed, so signs can alternate.
The implementation accumulates terms to avoid cancellation near a ratio of one. Results exceeding an absolute value of 10^18 are rejected; use an integer term count.
References:OpenStax: algebra and applications
What this helps you do
Find a final term and sum for an arithmetic or geometric sequence. Choose whether you repeatedly add a difference or multiply by a ratio.
What to enter
Start with sequence type, then fill the other relevant fields in the units shown. The field hints explain the required format and examples.
How the calculation works
Arithmetic term n is a1 + (n-1)*d and its sum is n*(a1+an)/2. Geometric term n is a1*r^(n-1); the sum is a1*(1-r^n)/(1-r), or n*a1 when r is one.
Worked example
Starting at 3 and adding 2 for four terms gives 3,5,7,9 with sum 24. Starting at 3 and multiplying by 2 gives 3,6,12,24 with sum 45.
Understanding the answer
A difference is an amount added each time; a ratio is a multiplier. Zero and negative geometric ratios are allowed, so signs can alternate.
Scope and limitations
The implementation accumulates terms to avoid cancellation near a ratio of one. Results exceeding an absolute value of 10^18 are rejected; use an integer term count.