Create an abstract representation of a geometric sequence. Show a sequence of shapes that increase in size according to an implied pattern. The 23rd shape should be comparatively smaller, drawn with dashed lines and colored in soft pink. The 28th shape should be slightly bigger, solid-lined, and colored in a faded purple hue. Then, to represent the 43rd term, include a significantly larger shape, in solid lines but with no color. The progression of sizes should visually indicate a steady rate of growth as per a geometric pattern. Note that there should be no text in the image.

The 23rd term in a certain geometric sequence is 16 and the 28th term in the sequence is 24. What is the 43rd term?

What is the answer?

I don't understand why so many people are getting 48, but that is not the answer. We know that in a geometric sequence, you multiply each term by a number to get your next term. So we know that we multiply our 23rd term by some number x 5 times to get our 28th number. So we know that 16 * x^5 = 24. All I did here was plug in the numbers. Therefore x^5 = 24/16 = 3/2, so x equals the fifth root of 3/2. We know that 24 * (fifth root of 3/2)^15 = T43 (T43 = 43rd term) . Simplifying this, we get T43 = 81.

81 is the correct answer, not 48

81

81

Let $r$ be the common ratio of the geometric sequence. To get from the $23^\text{rd}$ term of the sequence to the $28^\text{th}$ term, we start with the $23^{\text{rd}}$ term and multiply by $r$ five times. Therefore, we have

\[16r^5 = 24,\]so $r^5 = \frac{24}{16} = \frac32$. To get from the $28^\text{th}$ term to the $43^\text{rd}$ term, we start with the $28^\text{th}$ term and multiply by $r$ fifteen times. So, the $43^\text{rd}$ term is
\begin{align*}
24r^{15} &= 24\left(r^5\right)^3 \\
&= 24\left(\frac32\right)^3 \\
&= 24\cdot \frac{27}{8} \\
&= \boxed{81}.
\end{align*}

That is whyy the solution is 81.

T43 = T28 + 15(T28-T23)/(28-23)

= 24 + 15(24-16)/5 = 48