Permutations and Combinations

Permutations and Combinations

Question

Find the 2nd term and the rth term of an A.P. whose 6th term is 12 and 8th term is 22.

Answer

Let a be the first and d be the common difference
             straight t subscript 6 space equals space 12 space rightwards double arrow space straight a space plus space left parenthesis 6 space minus space 1 right parenthesis straight d space equals space 12 space rightwards double arrow space straight a space plus space 5 straight d space equals space 12              ...(i)
           space space space straight t subscript 8 space equals space 22 space rightwards double arrow space straight a space plus space left parenthesis 8 space minus space 1 right parenthesis straight d space equals space 22 space rightwards double arrow space straight a space plus space 7 straight d space equals space 22             ... (ii)
Subtracting (ii) from (i), we get
                         -2d = -10                   or                       d = 5
using value of d in (i), we get
                  a + 5(5) = 12                    or                        a = 12 - 25 = -13
                           straight t subscript 2 space equals space straight a space plus space left parenthesis 2 space minus space 1 right parenthesis straight d space equals space straight a space plus straight d space equals space minus 13 space plus space 5 space equals space minus 8
                          space space straight t subscript straight r space equals space straight a space plus space left parenthesis straight r minus 1 right parenthesis straight d space equals space minus 13 space plus space left parenthesis straight r space minus space 1 right parenthesis 5 space equals space minus 13 space plus space 5 straight r space minus space 5 space equals space 5 straight r space minus space 18

 

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