Question 5 and 6, Exercise 4.7

Solutions of Question 5 and 6 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Evaluate the sum: $\sum_{k=1}^{8} \frac{k}{k+1}$

Solution.

$$\begin{align*} \sum_{k=1}^{8} \frac{k}{k+1} &= \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\\ &+ \frac{6}{7} + \frac{7}{8} + \frac{8}{9} \\ &= 0.5 + 0.6667 + 0.75 + 0.8 + 0.8333\\ &+ 0.8571 + 0.875 + 0.8889 \\ &\approx 6.1709 \end{align*}$$

Evaluate the sum: $\sum_{k=1}^{7}(-1)^{k} 4^{k+1}$

Solution.

$$\begin{align*} \sum_{k=1}^{7} (-1)^{k} 4^{k+1} &= (-1)^1 \cdot 4^2 + (-1)^2 \cdot 4^3 + (-1)^3 \cdot 4^4 + (-1)^4 \cdot 4^5 \\ &\quad + (-1)^5 \cdot 4^6 + (-1)^6 \cdot 4^7 + (-1)^7 \cdot 4^8 \\ &= -16 + 64 - 256 + 1024 - 4096 + 16384 - 65536 \\ &= -49112 \end{align*}$$

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