Question

$open vertical bar table row cell 2 straight y plus 4 end cell cell space space space space 5 straight y plus 7 end cell cell space 8 straight y plus straight a end cell row cell 3 straight y plus 5 end cell cell space space space 6 straight y plus 8 end cell cell space space 9 straight y plus straight b end cell row cell 4 straight y plus 6 end cell cell space space space 7 straight y plus 9 end cell cell space space 10 straight y plus straight c end cell end table close vertical bar$ If a, b, c are in A.P. find value of

Solution

since space straight a comma straight b comma straight c space are space in space straight A. straight P therefore space space space space space space space 2 straight b equals straight a plus straight c Let space space space space increment space space equals space open vertical bar table row cell 2 straight y plus 4 end cell cell space space space space 5 straight y plus 7 end cell cell space 8 straight y plus straight a end cell row cell 3 straight y plus 5 end cell cell space space space 6 straight y plus 8 end cell cell space space 9 straight y plus straight b end cell row cell 4 straight y plus 6 end cell cell space space space 7 straight y plus 9 end cell cell space space 10 straight y plus straight c end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space space open vertical bar table row 0 0 cell straight a plus straight c minus 2 straight b end cell row cell 3 straight y plus 5 end cell cell space space space 6 straight y plus 8 end cell cell space space 9 straight y plus straight b end cell row cell 4 straight y plus 6 end cell cell space space space 7 straight y plus 9 end cell cell space space 10 straight y plus straight c end cell end table close vertical bar space. by space straight R subscript 1 rightwards arrow space straight R subscript 1 plus straight R subscript 3 space minus space 2 straight R subscript 2 space space space space space space space space space space space space space space space space equals space open vertical bar table row 0 0 cell straight a plus straight c minus 2 straight b end cell row cell 3 straight y plus 5 end cell cell space space space 6 straight y plus 8 end cell cell space space 9 straight y plus straight b end cell row cell 4 straight y plus 6 end cell cell space space space 7 straight y plus 9 end cell cell space space 10 straight y plus straight c end cell end table close vertical bar space space space space space open square brackets table row cell because space of space left parenthesis 1 right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space space space space space equals space 0 space space space space space space space space open square brackets table row cell because space each space elwment space of space straight R subscript 1 space is space zero end cell end table close square brackets space

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