The first term of an A.P is 5 and the common difference is -3/2, find the term whose value is -20
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The first term of an Arithmetic progression is 5 and the common difference is -3/2.find the term whose value is -41/2
An = 5 +(n-1) d
-20 = 5 + (n-1)(-1.5)
-25 = -1.5 n + 1.5
-26.5 = -1.5 n
n = 17 2/3
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Using the formula for the nth term of an arithmetic progression:
an = a1 + (n-1)d
where
an = term whose value is given = -41/2
a1 = first term of the arithmetic progression = 5
d = common difference = -3/2
-41/2 = 5 + (n-1)(-3/2)
-41/2 = 5 - (3/2)n + (3/2)
-46 = (-3/2)n
n = (46/3) + 1
Therefore, the term whose value is -41/2 is the (46/3) + 1th term of the arithmetic progression.
To find the term of an arithmetic progression (A.P), we can use the formula:
\[ {T_n} = a + (n - 1)d \]
where:
- \(T_n\) represents the \(n\)th term of the A.P.
- \(a\) is the first term of the A.P.
- \(d\) is the common difference between consecutive terms.
- \(n\) is the position of the term we want to find.
Given:
\(a = 5\)
\(d = -\frac{3}{2}\)
We need to find the term (\(T_n\)) whose value is -20.
Substituting the given values into the formula, we have:
\[ -20 = 5 + (n - 1) \left(-\frac{3}{2}\right) \]
Simplifying the equation, we get:
\[ -20 = 5 - \frac{3}{2}n + \frac{3}{2} \]
Combining like terms, we have:
\[ -20 - 5 - \frac{3}{2} = -\frac{3}{2}n \]
\[ -\frac{33}{2} = -\frac{3}{2}n \]
To solve for \(n\), we isolate the variable:
\[ n = \frac{-\frac{33}{2}}{-\frac{3}{2}} \]
Simplifying the expression:
\[ n = \frac{-33}{-3} \]
\[ n = 11 \]
Therefore, the term with a value of -20 is the 11th term of the arithmetic progression.