If the fifth term of a AP is 19 and the tenth term is 34 Find:- a) the first term b) The common difference c) The sum of the first 15 term

Answer:a) The first term is a=7b) The common difference is d=3c) The sum of the first 15 term is 420.     Step-by-step explanation:Given : If the fifth term of a AP is 19 and the tenth term is 34.To find : a) the first term b) The common difference c) The sum of the first 15 term ?Solution : The Arithmetic progression is in the form, [tex]a,a+d,a+2d,a+3d,...[/tex]Where, a is the first term and d is the common differenceThe nth term of the A.P is [tex]a_n=a+(n-1)d[/tex] The fifth term of a AP is 19.[tex]a_5=a+(5-1)d[/tex] [tex]19=a+4d[/tex] ...(1)The tenth term is 34.[tex]a_{10}=a+(10-1)d[/tex] [tex]34=a+9d[/tex] ...(2)Solving (1) and (2) by subtracting the equations,[tex]34-19=(a+9d)-(a+4d)[/tex][tex]15=a+9d-a-4d[/tex][tex]15=5d[/tex][tex]d=3[/tex]Substitute in (1),[tex]19=a+4(3)[/tex][tex]a=19-12[/tex][tex]a=7[/tex]a) The first term is a=7b) The common difference is d=3c) The sum of the first 15 term is given by, [tex]S_n=\frac{n}{2}[2a+(n-1)d][/tex][tex]S_{15}=\frac{15}{2}[2(7)+(15-1)3][/tex][tex]S_{15}=\frac{15}{2}[14+(14)3][/tex][tex]S_{15}=\frac{15}{2}[14+42][/tex][tex]S_{15}=\frac{15}{2}[56][/tex][tex]S_{15}=15\times 28[/tex][tex]S_{15}=420[/tex]The sum of the first 15 term is 420.