Arithmetic Series

 

Let a1, a2, a3, ..., a10, a11, a12 be an arithmetic sequence. If a1 + a2 = 4 and a2 + a3 = 5, then find a1.  

 

                                                         a₁ + a₂ = 4   &   a₂ + a₃ = 5  

 

Since it's an arithmetic sequence, the difference between adjacent terms is the same.  

 

Call that difference, i.e., the increment, let's call it "n".  Just keep this in mind for a while.  

 

                                                         a₂ + a₃ = 5    

                                                         a₁ + a₂ = 4    

Subtract one from the other  

                                                         a₃ – a₁ = 1  

Since a₃ is (a₁ + n + n)  

                                                         (a₁ + n + n) – a₁ = 1  

 

                                                          2n = 1  therefore n = 0.5  

Referring back to the top   

                                                          a₁ + a₂ = 4  

a₂ = (a₁ + 0.5) so plug it in  

                                                           a₁ + (a₁ + 0.5) = 4  

 

                                                           2a₁ + 0.5 = 4  

 

                                                           2a₁ = 4 – 0.5 = 3.5    

 

                                                            a₁ = 3.5 / 2  

 

                                                            a₁ = 1.75   

check answer  

                                                            a₁ = 1.75  

                                                            a₂ = 2.25        a₁ + a₂ = 4  

                                                            a₃ = 2.75        a₂ + a₃ = 5 

_.