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

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

T6=T4+2d, so 2d=12

Now you can find T1=a, and then T7

To find the first and seventh terms of the arithmetic progression (AP), we need to first determine the common difference (d) between the terms.

1. Using the given information, we know that the fourth term (a4) is 37, and the sixth term (a6) is 12 more than the fourth term, so a6 = a4 + 12.

2. The formula for the nth term of an arithmetic progression is: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.

3. Using the above formula, we can substitute n = 4 and n = 6 to form two equations:
a4 = a1 + 3d ---- (equation 1)
a6 = a1 + 5d ---- (equation 2)

4. Substituting the values we know, we can rewrite equation 2 as: a4 + 12 = a1 + 5d.

5. Now we have two equations with two unknowns (a1 and d):
a4 = a1 + 3d ---- (equation 1)
a4 + 12 = a1 + 5d ---- (equation 3)

6. Subtract equation 1 from equation 3 to eliminate a1:
(a4 + 12) - a4 = (a1 + 5d) - (a1 + 3d)
12 = 2d

7. We can divide both sides of the equation by 2, giving us d = 6.

8. Now that we have the value of d, we can substitute it back into equation 4 to solve for a4:
a4 = a1 + 3d
37 = a1 + 3(6)
37 = a1 + 18
a1 = 37 - 18
a1 = 19

9. We have found the value of a1, which is the first term of the arithmetic progression (AP). We can now use this value to find the seventh term (a7):
a7 = a1 + 6d
= 19 + 6(6)
= 19 + 36
= 55

Therefore, the first term (a1) of the AP is 19, and the seventh term (a7) is 55.