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10th Class Mathematics Arithmetic Progressions Question Bank Arithmetic Progressions

  • question_answer
    If \[x\ne y\] and the sequences x, \[{{a}_{1}},{{a}_{2}},y\] and x, \[{{b}_{1}},{{b}_{2}},\]y each are in A.P., then \[\left( \frac{{{a}_{2}}-{{a}_{1}}}{{{b}_{2}}-{{b}_{1}}} \right)\] is _____.

    A) \[\frac{2}{3}\]                        

    B) \[\frac{3}{2}\]            

    C) 1                     

    D)         \[\frac{3}{4}\]           

    Correct Answer: C

    Solution :

    For sequence, x, \[{{a}_{1}},{{a}_{2}},y\] \[y==x+3d\] \[\Rightarrow \] \[d=\frac{y-x}{3}\] \[\Rightarrow \] \[{{a}_{1}}=x+\frac{y-x}{3},{{a}_{2}}=x+2\left[ \frac{y-x}{3} \right]\] and Similarly,  \[{{a}_{2}}-{{a}_{1}}=\left[ \frac{y-x}{3} \right]\] For sequence, x,  \[{{b}_{1}},{{b}_{2}},y\] \[d'=\frac{y-x}{3}\] and \[{{b}_{2}}-{{b}_{1}}=\frac{y-x}{3}\] \[\Rightarrow \] \[\frac{{{a}_{2}}-{{a}_{1}}}{{{b}_{2}}-{{b}_{1}}}=1\]


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