The sum, $S_n$, of the first $n$ terms of a sequence $u_1, u_2, u_3, \dots$ is given by \(S_n

Question

The sum, $S_n$, of the first $n$ terms of a sequence $u_1, u_2, u_3, \dots$ is given by $S_n = 75 - \frac{3^{n+1}}{5^{n-2}}$. Show that $u_n = k\left(\frac{3}{5}\right)^{n-1}$, where $k$ is a constant to be determined. Find $S_\infty$.[4] Another sequence of numbers $v_1, v_2, v_3, \dots$ is such that $\sum_{i=1}^{n} v_i = \frac{2n}{2n + 1}$. Find the exact value of $\sum_{i=11}^{\infty} v_i$.[3]

Accepted Answer

(a) $k=30$, $u_n=30\left(\dfrac35\right)^{n-1}$, $S_\infty=75$ (b) $\dfrac1{21}$

2024 VJC Promo Q4

2024 VJC Promo Q4

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