Question

How do you find the first three terms of the arithmetic series \[n = 19\], \[{a_n} = 103\] and \[{S_n} = 1102\].

Answer 1

Hint:We can solve this using the formula of sum of all terms in a finite arithmetic progression with the last term given. That is \[{S_n} = \dfrac{n}{2}(a + l)\]. Using thus we find the first term. Then we find the common difference using the \[{n^{th}}\] formula that is \[{a_n} = a + (n - 1)d\]. Then we know that the general arithmetic progression form, \[a,a + d,a + 2d,a + 3d...........,a + (n - 1)d\].

Complete step by step solution:
Given,
\[n = 19\]. Then we have
\[{a_{19}} = 103\] and \[{S_{19}} = 1102\]
We have a sum formula that is \[{S_n} = \dfrac{n}{2}(a + l)\]. Where ‘a’ is first term, ‘l’ is the last term and ‘n’ is the position of the term.
Put \[n = 19\] in \[{S_n} = \dfrac{n}{2}(a + l)\]
\[{S_{19}} = \dfrac{{19}}{2}(a + l)\]
Here last term is \[{a_{19}} = 103(l)\] and substituting the given values we have,
\[1102 = \dfrac{{19}}{2}(a + 103)\]
Multiply by ‘2’ on both side we have,
\[1102 \times 2 = 19(a + 103)\]
\[1102 \times 2 = 19a + \left( {19 \times 103} \right)\]
\[2204 = 19a + 1957\]
\[ - 19a = - 2204 + 1957\]
\[ - 19a = - 247\]
Divide by -19 on both side we have,
\[a = \dfrac{{ - 247}}{{ - 19}}\]
\[ \Rightarrow a = 13\].
That is the first term is 13.
Now to find the common difference we have \[{a_n} = a + (n - 1)d\]. Where ‘a’ is the first term, ‘d’ is the common difference and ‘n’ is the position of the term.
Put \[n = 19\] in \[{a_n} = a + (n - 1)d\]
\[{a_{19}} = a + (19 - 1)d\]
\[{a_{19}} = a + 18d\]
Substituting the values we have
\[103 = 13 + 18d\]
\[103 - 13 = 18d\]
Simplifying and rearranging we have
\[18d = 90\]
Dividing by 18 on both sides,
\[d = \dfrac{{90}}{{18}}\]
\[ \Rightarrow d = 5\].
Thus the common difference is ‘5’.
Now from the general form of A.P is \[a,a + d,a + 2d,a + 3d...........,a + (n - 1)d\].
Thus the first three terms are \[a,a + d\] and \[a + 2d\]
\[a = 13\]
\[a + d = 13 + 5 = 18\]
\[a + 2d = 13 + 2(5) = 13 + 10 = 23\].
Hence the first three terms are 13,18 and 23.

Note: Remember all the formulas. We also know the sum of infinite terms formula in A.P is \[{S_n} = \dfrac{n}{2}(2a + (n - 1)d)\]. In arithmetic progression we find common differences but in geometric progression we find common ratios. The sum of infinite terms in geometric progression is \[{S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}\]. Where ‘r’ is the common ratio.