the fourth term of an ap is 37 and the sixth term is 12 more than the fourth find the first and seventh term
two differences (4th to 6th) is 12 ... d = 12 / 2 = 6
1st = 4th - 3 d
7th = 4th + 3 d
To find the first and seventh terms of an arithmetic progression (AP), we need to find the common difference (d) first.
We are given that the fourth term (a₄) is 37, and the sixth term (a₆) is 12 more than the fourth term. Hence, a₆ = 37 + 12 = 49.
Using the formula for the nth term of an AP, we have:
aₙ = a₁ + (n-1)d,
Where:
aₙ is the nth term,
a₁ is the first term, and
d is the common difference.
Let's substitute the given values:
a₄ = a₁ + (4-1)d,
37 = a₁ + 3d. - Equation (1)
a₆ = a₁ + (6-1)d,
49 = a₁ + 5d. - Equation (2)
Now, we can solve these two equations simultaneously to find the values of a₁ and d.
Subtracting equation (1) from equation (2), we get:
49 - 37 = a₁ + 5d - (a₁ + 3d),
12 = 5d - 3d,
12 = 2d.
Dividing both sides of the equation by 2, we have:
2d = 12,
d = 12/2,
d = 6.
Now, substitute the value of d back into equation (1):
37 = a₁ + 3(6),
37 = a₁ + 18.
Subtracting 18 from both sides:
19 = a₁.
So, the first term (a₁) is 19.
To find the seventh term (a₇), substitute the values of a₁ and d into the formula again:
a₇ = a₁ + (7-1)d,
a₇ = 19 + 6(7-1),
a₇ = 19 + 6(6),
a₇ = 19 + 36,
a₇ = 55.
Therefore, the first term (a₁) of the AP is 19, and the seventh term (a₇) is 55.