In the following APs, find the missing terms in the boxes : (i) 2, , 26 (ii) , 13, , 3 (iii) 5, , , 1 9 2 (iv) – 4, , , , , 6 (v) , 38, , , , – 22
Step-by-Step Solution
Key Concept: For an arithmetic progression (AP) with first term \(a_1\), common difference \(d\) and \(n\) terms, the \(k^{th}\) term is \(a_k = a_1 + (k-1)d\). If the first and last terms are known, \(d\) can be found using \(d = \dfrac{a_n - a_1}{n-1}\). Once \(d\) is known, any missing term can be obtained by adding appropriate multiples of \(d\) to the known term.
1. Identify the known terms and the total number of terms (\(n\)).\
2. Compute the common difference \(d\) using\
$$d = \frac{\text{last term} - \text{first term}}{n-1}.$$\
3. Use the formula\
$$a_k = a_1 + (k-1)d$$\
to find each missing term.\
(i) 2, __ , 26\
- Here \(n = 3\), \(a_1 = 2\), \(a_3 = 26\).\
- \(d = \dfrac{26-2}{2} = 12\).\
- Missing term \(a_2 = a_1 + d = 2 + 12 = 14\).\
(ii) __ , 13, __ , 3\
- \(n = 4\), \(a_2 = 13\), \(a_4 = 3\).\
- Let \(d\) be the common difference.\
- From \(a_4 = a_1 + 3d\) and \(a_2 = a_1 + d\):\
\( (a_1+3d) - (a_1+d) = 3-13 \Rightarrow 2d = -10 \Rightarrow d = -5\).\
- \(a_1 = a_2 - d = 13 - (-5) = 18\).\
- \(a_3 = a_2 + d = 13 + (-5) = 8\).\
(iii) 5, __ , __ , 192\
- \(n = 4\), \(a_1 = 5\), \(a_4 = 192\).\
- \(d = \dfrac{192-5}{3} = \dfrac{187}{3}\).\
- \(a_2 = a_1 + d = 5 + \dfrac{187}{3} = \dfrac{202}{3}\).\
- \(a_3 = a_2 + d = \dfrac{202}{3} + \dfrac{187}{3} = \dfrac{389}{3}\).\
(iv) –4, __ , __ , __ , __ , 6\
- \(n = 6\), \(a_1 = -4\), \(a_6 = 6\).\
- \(d = \dfrac{6-(-4)}{5} = \dfrac{10}{5} = 2\).\
- Successive terms: \(a_2 = -4+2 = -2\), \(a_3 = 0\), \(a_4 = 2\), \(a_5 = 4\).\
(v) __ , 38, __ , __ , __ , –22\
- \(n = 6\), \(a_2 = 38\), \(a_6 = -22\).\
- Let \(d\) be the common difference.\
- From \(a_6 = a_1 + 5d\) and \(a_2 = a_1 + d\):\
\((a_1+5d) - (a_1+d) = -22 - 38 \Rightarrow 4d = -60 \Rightarrow d = -15\).\
- \(a_1 = a_2 - d = 38 - (-15) = 53\).\
- \(a_3 = a_2 + d = 38 - 15 = 23\).\
- \(a_4 = a_3 + d = 23 - 15 = 8\).\
- \(a_5 = a_4 + d = 8 - 15 = -7\).
Correct Answer: (i) 14; (ii) 18, 13, 8, 3; (iii) \(\dfrac{202}{3}\) and \(\dfrac{389}{3}\) (i.e., 67\frac{1}{3} and 129\frac{2}{3}); (iv) -4, -2, 0, 2, 4, 6; (v) 53, 38, 23, 8, -7, -22