Planos

Calculadora de progressão aritmética (termo geral e soma)

Esta calculadora encontra o n-ésimo termo e a soma dos n primeiros termos de uma sequência que sobe ou desce sempre a mesma quantidade. Digite o primeiro termo, a razão e n.

Resultado

Digite os valores e toque em Calcular para ver o resultado.

Fórmula

an=a1+(n−1)da_n = a_1 + (n - 1)d

Start at the first term and add the difference n − 1 times.

Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)

The sum is the number of terms times the average of the first and last term.

Como usar esta calculadora

  1. Enter the first term and the common difference (the amount added each time; negative if the sequence goes down).
  2. Enter n, the term you want (and how many terms to add).
  3. Press Calculate to see the nth term, the sum and the first few terms.

Exemplos resolvidos

3, 7, 11, 15, … : the 10th term and the sum of 10 terms

a₁₀ = 3 + 9 × 4 = 39.

S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210.

A decreasing sequence: 20, 17, 14, …

a₈ = 20 + 7 × (−3) = −1.

S₈ = 8/2 × (20 + (−1)) = 4 × 19 = 76.

The sum 1 + 2 + … + 100

a₁₀₀ = 100, so S₁₀₀ = 100/2 × (1 + 100) = 5050.

Erros comuns

  • Using n instead of n − 1 in the nth-term formula: the first term already counts as one.
  • Getting the sign of d wrong for a decreasing sequence.
  • Adding the terms with the sum formula but the wrong last term.

Perguntas frequentes

How do you find the nth term of an arithmetic sequence?

Use aₙ = a₁ + (n − 1)d. For 3, 7, 11, …: a₁ = 3 and d = 4, so aₙ = 3 + 4(n − 1) = 4n − 1.

How do you find the common difference?

Subtract any term from the next one: 7 − 3 = 4. It is the same for every pair in an arithmetic sequence.

What is the difference between an arithmetic sequence and series?

A sequence is the list of terms (3, 7, 11, …); a series is their sum (3 + 7 + 11 + …).