Question 2, Exercise 4.2

Solutions of Question 2 of Exercise 4.2 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.

Find the next three terms of each arithmetic sequence. $5,9,13, \ldots$

Solution.

Give: $$5, 9, 13, \ldots $$ Thus $a_1=5$, $d=9-5=4$.
Now $$a_n=a_1+(n-1)d.$$ So, we have \begin{align*} a_4 &=5+(4-1)(4)=5+12=17\\ a_5 &=5+(5-1)(4)=5+16=21\\ a_6 &=5+(6-1)(4)=5+20=25 \end{align*} Thus, the next three terms of the sequence are $17$, $21$, $25$.

Find the next three terms of each arithmetic sequence. $11,14,17, \ldots$

Solution.

Given: $$11, 14, 17, \ldots$$ Thus $a_1=11$, $d=14-11=3$.
Now $$a_n=a_1+(n-1)d.$$ So, we have \begin{align*} a_4 &= 11 + (4-1) \cdot 3 = 11 + 9 = 20\\ a_5 &= 11 + (5-1) \cdot 3 = 11 + 12 = 23\\ a_6 &= 11 + (6-1) \cdot 3 = 11 + 15 = 26 \end{align*} Thus, the next three terms of the sequence are $20$, $23$, $26$.

Find the next three terms of each arithmetic sequence. $\dfrac{1}{2}, \dfrac{3}{2}, \dfrac{5}{2}, \ldots$

Solution.

The given sequence is $$\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots$$ First, we find the common difference: \begin{align*} d &= \frac{3}{2} - \frac{1}{2} = 1 \end{align*} The arithmetic sequence is: $$a_n = a_{n-1} + d$$ Now, let's calculate the next three terms: \begin{align*} a_3 &= \frac{5}{2}\\ a_4 &= a_3 + d = \frac{5}{2} + 1 = \frac{7}{2}\\ a_5 &= a_4 + d = \frac{7}{2} + 1 = \frac{9}{2}\\ a_6 &= a_5 + d = \frac{9}{2} + 1 = \frac{11}{2} \end{align*} Thus, the next three terms of the sequence are $ \dfrac{7}{2}, \dfrac{9}{2}, \dfrac{11}{2} $.

Find the next three terms of the arithmetic sequence. $-5.4,-1.4,-2.6, \ldots$

Solution.

Given: $$-5.4, -1.4, 2.6, \ldots$$ Thus, \(a_1 = -5.4\), and \(d = -1.4 - (-5.4) = 4\). Now $$a_n = a_1 + (n - 1)d.$$ So, we have \[ \begin{aligned} a_4 &= -5.4 + (4 - 1)(4) = -5.4 + 12 = 6.6, \\ a_5 &= -5.4 + (5 - 1)(4) = -5.4 + 16 = 10.6, \\ a_6 &= -5.4 + (6 - 1)(4) = -5.4 + 20 = 14.6. \end{aligned} \] Thus, the next three terms of the sequence are \(6.6\), \(10.6\), and \(14.6\).

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