Question 29

In an increasing geometric progression with $$1st$$ term $$a$$ and $$n$$th term $$t_n$$, the difference between the fourth and first terms is $$52$$ and the sum of the first three terms is $$26$$. Then the numerical value of $$\frac{t_{2024}}{t_{2023}}+\frac{a^{2024}}{a^{2023}}$$ is

Correct Answer: 5

Solution

If the common ratio is $$r$$, then $$a(r^3-1)=52$$ and $$a(1+r+r^2)=26$$. Since $$r^3-1=(r-1)(1+r+r^2)$$, division gives $$r-1=2$$, so $$r=3$$. Substitution gives $$a=2$$. Therefore $$\frac{t_{2024}}{t_{2023}}+\frac{a^{2024}}{a^{2023}}=r+a=3+2=5$$.

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