the Fourth term of an a.p is 37 and the 6th term is 12 more than the Fourth term.find the first and seventh terms

Answer 1

T6=T4+2d, so 2d=12
Now you can find T1=a, and then T7

Answer 2

a4 = 37

a6 = a4 + 12

a6 = 37 + 12 = 49

In an arithmetic progression nth term of the sequence (an) is given by:

an = a1 + ( n - 1 ) d

so:

a4 = a1 + ( 4 - 1 ) d = 37

a1 + 3 d = 37

a6 = a1 + ( 6 - 1 ) d = 49

a1 + 5 d = 49

Now you must solve system of two equations with two unknowns :

a1 + 3 d = 37

a1 + 5 d = 49

a1 + 3 d = 37 Subtract 3 d to both sides

a1 + 3 d - 3 d = 37 - 3 d

a1 = 37 - 3 d

a1 + 5 d = 49

37 - 3 d + 5 d = 49

37 + 2 d = 49 Subtract 37 to both sides

37 + 2 d - 37 = 49 - 37

2 d = 12 Divide both sides by 2

d = 6

a1 + 5 d = 49

a1 + 5 * 6 = 49

a1 + 30 = 49 Subtract 30 to both sides

a1 + 30 - 30 = 49 - 30

a1 = 19

a7 = a1 + ( 7 - 1 ) d

a7 = 19 + 6 d

a7 = 19 + 6 * 6

a7 = 19 + 36

a7 = 55