the first term of an arithmetic progression is-4 and 15th term is double the 5th term. find the 12th term.
Given F1= -4
So F15= F1+14d=-4+14d=2[F1+4d]=2F1+8d
-4+14d=2*-4+8d=-8+8d
14d-8d=-8+4
6d=-4 d=-4/6=-2/3
F12=F1+11d=-4+11*2/3=-4+22/3=-12+22/3=10/3=3 1/3
search for Math is Fun arithmetic sequence
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https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
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To find the 12th term of the arithmetic progression, we first need to determine the common difference (d) of the sequence.
Let's denote the first term of the arithmetic progression as a₁, the 5th term as a₅, and the 15th term as a₁₅.
Given:
a₁ = -4
a₁₅ = 2(a₅)
We can use the formula for the nth term of an arithmetic progression to establish a relationship:
aₙ = a₁ + (n - 1)d
Now, we can use this formula to find the common difference:
a₁₅ = a₁ + (15 - 1)d
Substituting the given values:
2(a₅) = -4 + (15 - 1)d
Since a₅ is not explicitly given, we need to express it in terms of a₁ and d. We can use the same formula again for the 5th term:
a₅ = a₁ + (5 - 1)d
Substituting the given value of a₁:
a₅ = -4 + (5 - 1)d
Now, we can substitute this expression for a₅ back into the equation aₙ = 2(a₅):
2(-4 + (5 - 1)d) = -4 + (15 - 1)d
Simplifying the equation:
-8 + 8d = -4 + 15d
Bringing like terms together:
-8 + 4 = 15d - 8d
-4 = 7d
Solving for d:
d = -4/7
Now that we have the value of the common difference, we can find the 12th term (a₁₂) using the same formula:
a₁₂ = a₁ + (12 - 1)d
Substituting the given values:
a₁₂ = -4 + (12 - 1)(-4/7)
Simplifying the expression:
a₁₂ = -4 + (11)(-4/7)
a₁₂ = -4 + (-44/7)
a₁₂ = -4 - 44/7
To combine fractions with different denominators, we need a common denominator:
-4 = -28/7
a₁₂ = (-28 - 44)/7
a₁₂ = -72/7
Thus, the 12th term of the arithmetic progression is -72/7, or approximately -10.2857.