The first term of an Arithmetic Progression is -4 and the 15th term is double the 5th.

Find the 12th term of the AP

Just use your definitions:

"first term of an Arithmetic Progression is -4"
--> a=-4

"the 15th term is double the 5th" ---> a+14d = 2(a+4d)
a + 14d = 2a + 8d
6d = a
6d = -4
d = -2/3

term 12 = a + 11d
= -4 + 11(-2/3)
= -25/3

check: is the 15th term twice the 5th ??
term 15 = -4+14(-2/3) = -40/3
term 5 = -4 + 4(-2/3) = -20/3
YES

-4+11(-2/3) will not give 25/3 but 34/3. Check that.

To find the 12th term of an arithmetic progression (AP), we need to know the common difference (d) between the terms.

Given:
First term (a₁) = -4
15th term (a₁₅) = 2 * 5th term (a₅)

Step 1: Find the common difference (d)
We can find the common difference by subtracting the first term (a₁) from the 5th term (a₅).
d = a₅ - a₁

Step 2: Find the 5th term (a₅)
We can find the 5th term by expressing it in terms of the first term and common difference.
a₅ = a₁ + 4d

Step 3: Find the value of d
We know that 15th term (a₁₅) is double the 5th term (a₅).
a₁₅ = 2 * a₅
Substitute the value of a₅ that we found in step 2 and solve for d:
-4 + 14d = 2 * (-4 + 4d)

Step 4: Solve for d
Simplify the equation:
-4 + 14d = -8 + 8d

Combine like terms:
14d - 8d = -8 + 4

Solve for d:
6d = -4

Divide both sides by 6:
d = -4/6
= -2/3

Step 5: Find the 12th term (a₁₂)
Using the formula for the nth term of an AP:
aₙ = a₁ + (n - 1) * d

Substitute the values we have:
a₁₂ = -4 + (12 - 1) * (-2/3)
= -4 + 11 * (-2/3)
= -4 + (-22/3)
= (-12/3) + (-22/3)
= -34/3

Therefore, the 12th term of the AP is -34/3.

To find the 12th term of the arithmetic progression (AP), we need to figure out the common difference of the AP first.

Let's denote the first term of the AP as 'a', and the common difference as 'd'.

Given that the first term of the AP is -4, we have:
a = -4

Also, it is given that the 15th term is double the 5th term. So, we can set up the following equation:
a + 14d = 2(a + 4d)

Substituting the value of 'a':
-4 + 14d = 2(-4 + 4d)

Simplifying this equation, we get:
-4 + 14d = -8 + 8d

Bringing all the 'd' terms to one side and all constant terms to the other side:
14d - 8d = -8 + 4

Simplifying further:
6d = -4

Finally, solving for 'd':
d = -4/6

Simplifying the fraction:
d = -2/3

Now that we know the value of the common difference 'd', we can find the 12th term of the AP using the formula:
tn = a + (n - 1)d

Substituting the values:
t12 = -4 + (12 - 1) * (-2/3)

Simplifying the expression:
t12 = -4 + 11 * (-2/3)

To calculate the answer, do the following steps:
1. Multiply the denominator '3' with the whole number '11' in the numerator, which gives -22.
2. Add -22 and -4, which equals -26.
3. Divide -26 by the denominator '3', which gives -8.6666667 (approximately).

Therefore, the 12th term of the AP is approximately -8.6666667.