Question
The 9th and 12th term of the A.p are 50and 65 respectivly. Fine (a) the common difference (b) the sum of its first 70 terms
Answers
Bosnian
In an Arithmetic progression:
an = a1 + ( n - 1 ) d
where
a1 = the initial term
an = the nth term
d = the common difference of successive members
In this case:
a9 = a1 + 8 d = 50
a12 = a1 + 11 d = 65
Now you must solve system of two equations:
a1 + 8 d = 50
a1 + 11 d = 65
The solutions are:
a1 = 10 , d = 5
The sum of its first n terms:
Sn = n ( a1 + an ) / 2
In this case:
a70 = a1 + 69 d = 10 + 69 ∙ 5 = 355
n = 70
Sn = n ( a1 + an ) / 2
S70 = 70 ( a1 + a70 ) / 2
S70 = 70 ( 10 + 355 ) / 2 = 12775
The sum of its first n terms also can calculate by formula:
Sn = n [ 2 a1 + ( n - 1 ) d ] / 2
S70 = 70 ( 2 ∙10 + 69 ∙ 5 ) / 2 = 12775
an = a1 + ( n - 1 ) d
where
a1 = the initial term
an = the nth term
d = the common difference of successive members
In this case:
a9 = a1 + 8 d = 50
a12 = a1 + 11 d = 65
Now you must solve system of two equations:
a1 + 8 d = 50
a1 + 11 d = 65
The solutions are:
a1 = 10 , d = 5
The sum of its first n terms:
Sn = n ( a1 + an ) / 2
In this case:
a70 = a1 + 69 d = 10 + 69 ∙ 5 = 355
n = 70
Sn = n ( a1 + an ) / 2
S70 = 70 ( a1 + a70 ) / 2
S70 = 70 ( 10 + 355 ) / 2 = 12775
The sum of its first n terms also can calculate by formula:
Sn = n [ 2 a1 + ( n - 1 ) d ] / 2
S70 = 70 ( 2 ∙10 + 69 ∙ 5 ) / 2 = 12775