If the 3rd term of a geometric sequence is 12 and the common ratio is 2, what is

If the 3rd term of a geometric sequence is 12 and the common ratio is 2, what is the first term? (2023)

If the 3rd term of a geometric sequence is 12 and the common ratio is 2, what is the first term? (2023)

Step 1: Understand that in a geometric sequence, each term is found by multiplying the previous term by a common ratio.
Step 2: Identify the formula for the nth term of a geometric sequence, which is given by T_n = a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
Step 3: Since we are looking for the 3rd term (n=3), we can write the formula as T_3 = a * r^(3-1) = a * r^2.
Step 4: We know the 3rd term (T_3) is 12 and the common ratio (r) is 2. So we can substitute these values into the formula: 12 = a * (2^2).
Step 5: Calculate 2^2, which is 4. Now the equation is 12 = a * 4.
Step 6: To find 'a', divide both sides of the equation by 4: a = 12 / 4.
Step 7: Calculate 12 divided by 4, which equals 3. Therefore, the first term 'a' is 3.

Geometric Sequence – A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Term Calculation – Understanding how to calculate specific terms in a geometric sequence using the formula for the nth term.