If the first term of an AP is 5 and it's 100th term is -292.find it's 50th term.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence an is given by:
an = a1 + ( n -1 ) * d
a1 = 5
a100 = a1 + ( 100 - 1 ) * d = - 292
a100 = 5 + 99 d = - 292
5 + 99 d = - 292 Subtract 5 to both sides
5 + 99 d - 5 = - 292 - 5
99 d = - 297 Divide both sides by 99
d = - 297 / 99 = -3
d = - 3
an = a1 + ( n -1 ) * d
a50 = 5 + ( 50 -1 ) * d
a50 = 5 + 49 * d
a50 = 5 + 49 * ( - 3 )
a50 = 5 - 147
a50 = - 142
Halifax | Managing Your Money | Money Worries
To find the 50th term of an arithmetic progression (AP) given the first term and the 100th term, we need to find the common difference (d) first.
The formula for the nth term of an AP is given by:
Tn = a + (n - 1)d
Where:
Tn = nth term
a = first term
n = position/index of the term
d = common difference
Using the known values, we can substitute the given information into the formula:
T100 = a + (100 - 1)d
-292 = 5 + 99d (substituting T100 = -292, a = 5, and n = 100)
Now, we can solve this equation to find the value of d:
-292 - 5 = 99d
-297 = 99d
d = -3 (dividing both sides by 99)
Now that we have the common difference (d = -3), we can find the 50th term (T50):
T50 = a + (50 - 1)d (substituting a = 5, n = 50, and d = -3)
T50 = 5 + (49)(-3)
T50 = 5 - 147
T50 = -142
Therefore, the 50th term of the arithmetic progression is -142.
To find the 50th term of an arithmetic progression (AP), we need to have the common difference (d) between consecutive terms. With only the first term (a₁) and the 100th term (a₁₀₀) given, we can find the common difference using the formula for the nth term of an AP:
aₙ = a₁ + (n - 1)d
Given:
a₁ = 5
a₁₀₀ = -292
Let's calculate the common difference (d) first using the given values:
-292 = 5 + (100 - 1)d
Simplifying the equation:
-292 = 5 + 99d
Rearranging the equation:
99d = -297
Dividing both sides by 99:
d = -3
Now that we have the common difference (d = -3), we can use the first term (a₁) and the common difference (d) to find the 50th term (a₅₀):
a₅₀ = a₁ + (50 - 1)d
Plugging in the values:
a₅₀ = 5 + (50 - 1)(-3)
Simplifying the equation:
a₅₀ = -139