Geometric Sequence: nth Term (Notes and Examples)

Definitions and Key Formula

  • Geometric sequence: a sequence where the ratio between consecutive terms is constant.

  • Variables in the nth-term formula:

    • a_n = the nth term of the sequence.

    • a_1 = the first term of the sequence.

    • r = the common ratio (the factor between successive terms).

    • n = the term position (the term number you want).

  • Core formula for the nth term:

    • an = a1 rn-1

    • Explanation: exponent is (n-1) because when (n=1) you should get the first term: (a1 = a1) and ( an = the nth term)

How to identify geometric vs arithmetic sequences

  • Look at consecutive terms:

    • Arithmetic sequence has a constant difference: (d = a

    • Geometric sequence has a constant ratio: (r = \dfrac{a{k+1}}{ak}).

  • Example sequence: 3, 6, 12, 24

    • Check differences: (6-3=3, 12-6=6, 24-12=12) — not constant, so not arithmetic.

    • Check ratios: (\dfrac{6}{3}=2, \dfrac{12}{6}=2, \dfrac{24}{12}=2) — constant ratio, so geometric with r=2.

  • Identify initial term: a_1 = 3.

Step-by-step method to find a_n for a geometric sequence

  • Step 1: Verify the sequence is geometric (find and confirm the common ratio r).

  • Step 2: Record the known values from the sequence: a_1 and r.

  • Step 3: Decide which term you need, i.e., determine n.

  • Step 4: Substitute into the nth-term formula: an = a1 \, r^{\,n-1}.

  • Step 5: Compute the power and then multiply by the first term.

  • Note on computation:

    • Powers can be computed by hand or with a calculator (e.g., 2^{6} = 64, \; 2^{9} = 512).

Example 1: Find the seventh term of 3, 6, 12, 24

  • Given sequence: 3, 6, 12, 24

  • Determine parameters:

    • a_1 = 3

    • Common ratio: r = \dfrac{6}{3} = 2 (also \dfrac{12}{6}=2, \dfrac{24}{12}=2)

  • Target term: n=7

  • Compute using the formula:

    • a7 = a1 \cdot r^{7-1} = 3 \cdot 2^{6}

    • Evaluate: 2^{6} = 64

    • Multiply: 3 \cdot 64 = 192

  • Result: a_7 = 192

  • Interpretation: The seventh term of the sequence is 192.

Example 2: Find the tenth term of 3, 6, 12, 24

  • Given sequence: 3, 6, 12, 24 (same as above)

  • Parameters unchanged:

    • a_1 = 3

    • r = 2

  • Target term: n=10

  • Compute using the formula:

    • a{10} = a1 \cdot r^{10-1} = 3 \cdot 2^{9}

    • Evaluate: 2^{9} = 512

    • Multiply: 3 \cdot 512 = 1536

  • Result: a_{10} = 1536

  • Interpretation: The tenth term of the sequence is 1536.

Quick assignment (from the video)

  • Using the same sequence, find the 12th term:a{12} = a1 \cdot r^{12-1} = 3 \cdot 2^{11} = 3 \cdot 2048 = 6144.

  • Note: This follows directly from the same steps; the 12th term would be 6144 given the sequence 3, 6, 12, 24.

Practical tips and common pitfalls

  • Always confirm the type of sequence before applying the formula.

  • Be careful with the exponent: it is always (n-1) for the nth term formula.

  • If a term or ratio is negative or fractional, the same method applies: compute with the given a_1 and r.

  • Use a calculator for large powers, but verify smaller powers with mental math where convenient.

  • Remember the interpretation: geometric sequences exhibit exponential growth or decay depending on the sign and magnitude of r.

Connections to broader concepts

  • The nth-term formula is a specific case of exponential functions: terms grow by a constant factor each step.

  • Real-world relevance includes compound interest, population growth/decay, and any process with multiplicative changes over time.

  • Foundational principle: discrete exponential dynamics, in contrast to linear changes in arithmetic sequences.

Summary of key formulas

  • Geometric nth term: an = a1 \, r^{\,n-1}

  • Common ratio from terms: r = \dfrac{a{k+1}}{ak} \, (k\ge 1)

  • Example computations from the transcript:

    • For the sequence 3, 6, 12, 24 with a_1 = 3, \; r = 2:

    • a_7 = 3 \cdot 2^{6} = 192

    • a_{10} = 3 \cdot 2^{9} = 1536