The twenty-first term of an Arithmetic Progression is \(5\frac{1}{2}\) and the sum of the first twenty-one terms is \(94\frac{1}{2}\). Find the : (a) first ...
Question 1 Report
The twenty-first term of an Arithmetic Progression is \(5\frac{1}{2}\) and the sum of the first twenty-one terms is \(94\frac{1}{2}\). Find the :
(a) first term ; (b) common difference ; (c) sum of the first thirty terms.
Answer Details
A.P.: \(T_{21}=5\tfrac12\) and \(S_{21}=94\tfrac12\).
Write the two facts algebraically:
\[T_{21}=a+20d=\frac{11}{2}\qquad(1)\]\[S_{21}=\frac{21}{2}\big(2a+20d\big)=\frac{189}{2}\;\Rightarrow\;2a+20d=9\;\Rightarrow\;a+10d=\frac{9}{2}\qquad(2)\]Subtract (2) from (1):
\[10d=\frac{11}{2}-\frac{9}{2}=1\;\Rightarrow\;d=\frac{1}{10}\]From (2), \(a=\dfrac92-10\left(\tfrac{1}{10}\right)=\dfrac92-1=\dfrac72\).
(a) First term \(a=\dfrac72=3\tfrac12\).
(b) Common difference \(d=\dfrac{1}{10}=0.1\).
(c) Sum of the first 30 terms:
\[S_{30}=\frac{30}{2}\big(2a+29d\big)=15\left(7+29\times 0.1\right)=15(7+2.9)=15(9.9)=148.5\]So \(S_{30}=148\tfrac12\).
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