the fourth terms of an AP is 37 and the 6th terms is 12 more than the fourth term.Find the first and seventh terms.
clearly d=6, since a6 = a4+12 = a4+2d
so, since a4=a+3d=a+18=37, a=19
a1=a=19
a7=a+6d=19+6*6=44
To find the first and seventh terms of an arithmetic progression (AP), we first need to find the common difference (d).
Given that the fourth term (a₄) is 37, we can use the formula for the nth term of an AP:
aₙ = a₁ + (n - 1) * d
Substituting n = 4 and a₄ = 37 into the formula, we have:
37 = a₁ + (4 - 1) * d
Now, let's find the value of d.
37 = a₁ + 3d
Next, we are given that the 6th term (a₆) is 12 more than the fourth term (a₄).
a₆ = a₄ + 12
Substituting the values, we have:
a₁ + 5d = a₁ + 3d + 12
Simplifying the equation, we get:
2d = 12
d = 6
Now that we have the common difference (d = 6), we can find the first term (a₁).
Substituting the values, we have:
37 = a₁ + 3 * 6
37 = a₁ + 18
a₁ = 37 - 18
a₁ = 19
Therefore, the first term (a₁) is 19.
To find the seventh term (a₇), we can use the formula for the nth term of an AP:
aₙ = a₁ + (n - 1) * d
Substituting n = 7, a₁ = 19, and d = 6 into the formula, we have:
a₇ = 19 + (7 - 1) * 6
a₇ = 19 + 6 * 6
a₇ = 19 + 36
a₇ = 55
Therefore, the seventh term (a₇) is 55.