Question 21 and 22, Exercise 4.1

Solutions of Question 21 and 22 of Exercise 4.1 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.

Predict the general term or nth term, $a_{n}$, of the sequence. $\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$

Solution.

The given sequence is $$\sqrt{2}, \sqrt{4}, \sqrt{6}, \sqrt{8}, \sqrt{10}, \ldots$$ The terms can be rewritten as: \begin{align*} &a_1=\sqrt{2 \cdot 1}, \\ &a_2=\sqrt{4}=\sqrt{2 \cdot 2} \\ &a_3=\sqrt{6}=\sqrt{2 \cdot 3}\\ &a_4=\sqrt{8}=\sqrt{2 \cdot 4}\\ &a_5=\sqrt{10}=\sqrt{2 \cdot 5} \end{align*} Thus, we can predict $a_n=\sqrt{2n}$.

Predict the general term or nth term, $a_{n}$, of the sequence. $1.2,2.3,3.4,4.5, \ldots$

Solution. The given sequence is in arithmetic in which $a_1=1.2$, $d=2.3-1.2=1.1$, thus \begin{align*} a_n &= a + (n - 1) d\\ &=1.2+(n-1)(1.1)\\ &=1.2+1.1n-1.1\\ &=1.1n+0.1 \end{align*} Thus the general term is $a_n=1.1n+0.1$.

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