find the nth term of the sequence 3,9,27,81
Answer (1)
Answer:Step 1: Analyze the SequenceFirst, look at the relationship between the numbers.The 1st term is 3The 2nd term is 9 (which is 3 x 3)The 3rd term is 27 (which is 9 x 3)The 4th term is 81 (which is 27 x 3)We can see that each term is found by multiplying the previous term by 3. This means it is a geometric sequence with a common ratio (r) of 3.Step 2: Use the Formula for a Geometric SequenceThe standard formula for the nth term (aₙ) of a geometric sequence is:aₙ = a₁ * rⁿ⁻¹Where:aₙ is the nth term (what we want to find)a₁ is the first term of the sequencer is the common ration is the term numberStep 3: Substitute the Values from Your SequenceFrom our sequence (3, 9, 27, 81):The first term (a₁) = 3The common ratio (r) = 3Now, substitute these values into the formula:aₙ = 3 * 3ⁿ⁻¹Step 4: Simplify the ExpressionTo simplify 3 * 3ⁿ⁻¹, we use the rule of exponents which states that xᵃ * xᵇ = xᵃ⁺ᵇ.In our case, 3 is the same as 3¹.So, we have:aₙ = 3¹ * 3ⁿ⁻¹aₙ = 3¹⁺⁽ⁿ⁻¹⁾aₙ = 3ⁿVerificationLet's test our formula aₙ = 3ⁿ to make sure it works:For the 1st term (n=1): 3¹ = 3 (Correct)For the 2nd term (n=2): 3² = 9 (Correct)For the 3rd term (n=3): 3³ = 27 (Correct)For the 4th term (n=4): 3⁴ = 81 (Correct)The formula is correct.
