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  2. AP Pre Calculus
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Arithmetic sequence: Formula for the nth term (given initial term).

an=a0+dna_n = a_0 + dnan​=a0​+dn

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Arithmetic sequence: Formula for the nth term (given initial term).

an=a0+dna_n = a_0 + dnan​=a0​+dn

Arithmetic sequence: Formula for the nth term (given a known term).

an=ak+d(n−k)a_n = a_k + d(n-k)an​=ak​+d(n−k)

Geometric sequence: Formula for the nth term (given initial term).

gn=g0∗rng_n = g_0 * r^ngn​=g0​∗rn

Geometric sequence: Formula for the nth term (given a known term).

gn=gk∗r(n−k)g_n = g_k * r^(n-k)gn​=gk​∗r(n−k)

Arithmetic vs. Geometric: What is the key difference in how terms change?

Arithmetic: Constant addition/subtraction (common difference) | Geometric: Constant multiplication/division (common ratio)

Compare the growth patterns of arithmetic and geometric sequences.

Arithmetic: Linear growth | Geometric: Exponential growth

How to find the 15th term of an arithmetic sequence given a_0 and d?

Use the formula an=a0+dna_n = a_0 + dnan​=a0​+dn. Substitute n = 15, a_0 with the initial term, and d with the common difference.

How to find the 9th term of a geometric sequence given g_0 and r?

Use the formula gn=g0∗rng_n = g_0 * r^ngn​=g0​∗rn. Substitute n = 9, g_0 with the initial term, and r with the common ratio.

How to determine if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. If it is, the sequence is arithmetic.

How to determine if a sequence is geometric?

Check if the ratio between consecutive terms is constant. If it is, the sequence is geometric.

Given two terms in an arithmetic sequence, how do you find the common difference?

Use the formula an=ak+d(n−k)a_n = a_k + d(n-k)an​=ak​+d(n−k) and solve for d.

Given two terms in a geometric sequence, how do you find the common ratio?

Use the formula gn=gk∗r(n−k)g_n = g_k * r^(n-k)gn​=gk​∗r(n−k) and solve for r.