The first term of an ap is 6 and the fifth term is 18 find the number of series having a sim of 162
d=3, so you want n such that
n/2 (12+3(n-1)) = 162
To find the number of terms in an arithmetic progression (AP) with a given sum, we need to first find the common difference (d) between terms.
Given that the first term (a₁) is 6 and the fifth term (a₅) is 18, we can find the common difference by using the formula:
a₅ = a₁ + (n-1)d
where n is the number of terms.
Plugging in the given values, we have:
18 = 6 + 4d
Subtracting 6 from both sides, we get:
12 = 4d
Dividing both sides by 4, we find that the common difference (d) is 3.
Now, we can find the number of terms using the sum formula for an AP:
S = (n/2) * (2a₁ + (n-1)d)
Given that the sum (S) is 162, a₁ is 6, and d is 3, we can substitute these values into the formula:
162 = (n/2) * (2 * 6 + (n-1) * 3)
Expanding and simplifying:
162 = 3n² + 3n
Rearranging the equation and converting it to a quadratic form:
3n² + 3n - 162 = 0
Now, we can solve this quadratic equation to find the value(s) of n using factoring, completing the square, or the quadratic formula. Once we find the solutions, we note that the discriminant (b² - 4ac) should be a perfect square for us to have a whole number solution.
To find the number of terms in the series that has a sum of 162, we need to determine the common difference of the arithmetic progression (AP) first.
Given:
First term (a₁) = 6
Fifth term (a₅) = 18
We can use the following formula to find the nth term of an arithmetic progression:
aₙ = a₁ + (n - 1)d
Using the given information:
a₁ = 6
a₅ = 18
To find the common difference (d), substitute the values of a₁ and a₅ into the formula:
18 = 6 + (5 - 1)d
Simplify the equation:
18 = 6 + 4d
Move 6 to the other side:
18 - 6 = 4d
12 = 4d
Divide both sides by 4:
12/4 = 4d/4
3 = d
So, the common difference (d) of the arithmetic progression is 3.
To find the number of terms (n) that have a sum of 162, we can use the formula for the sum of an arithmetic progression:
Sₙ = (n/2)(2a₁ + (n - 1)d)
Substitute the known values:
162 = (n/2)(2(6) + (n - 1)3)
Simplify the equation:
162 = (n/2)(12 + 3(n - 1))
Distribute:
162 = (n/2)(12 + 3n - 3)
Combine like terms:
162 = (n/2)(9 + 3n)
Multiply both sides by 2:
2(162) = n(9 + 3n)
324 = n(9 + 3n)
Rearrange the equation:
3n² + 9n - 324 = 0
Factorize or use the quadratic formula to solve the equation for n.
Factoring:
(3n - 18)(n + 18) = 0
Therefore, n = 18 or n = -18
Since the series cannot have a negative number of terms, we consider n = 18 as the solution.
Thus, the number of terms required to have a sum of 162 is 18 in the given arithmetic progression.