Question

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Ex1: a1=7, r=5, S9=?

Ex2: a10=19, S10=100, a1=?, r=?

Answer 1

$\displaystyle 1).a_1=7,r=5,S_9=? \\ S_n= \frac{2a_1+(n-1) \cdot 3}{2} \cdot n \\ S_9= \frac{2 \cdot 7+(9-1) \cdot 5}{2} \cdot 9 \\ S_9= \frac{14+8 \cdot 5}{2} \cdot 9 \\ S_9= \frac{14+40}{2} \cdot 9 \\ S_9= \frac{54}{2} \cdot 9 \\ S_9=27 \cdot 9 \\ S_9=243$
$\displaystyle 2).a_{10}=19,S_{10}=100,a_1=?,r=? \\ S_{10}= \frac{a_1+a_{10}}{2} \cdot 10 \\ 100=(a_1+19) \cdot 5 \\ 100=5a_1+95 \\ 5a_1=100-95 \\ 5a_1=5 \\ a_1= \frac{5}{5} \\ a_1=1 \\ a_{10}=19 \Rightarrow a_{10-1}+r=19 \Rightarrow a_9+r=19 \Rightarrow a_1+9r=19 \Rightarrow \\ \Rightarrow 1+9 r=19 \Rightarrow 9r=19-1 \Rightarrow 9r=18 \Rightarrow r= \frac{18}{9} \Rightarrow r=2$