The sum of the first In this of the son as4+7+10+......is 20, find n
We can start by finding a formula for the nth term in the sequence 4, 7, 10, ... . We can see that each term is 3 more than the previous term, so we can write:
a_n = a_1 + (n-1)d
where a_1 = 4, d = 3, and n is the number of terms we want to find. Substituting these values, we get:
a_n = 4 + 3(n-1)
a_n = 3n + 1
Now we can use the formula for the sum of an arithmetic series:
S_n = (n/2)(a_1 + a_n)
We know that the sum of the first n terms is 20, so we can write:
20 = (n/2)(4 + 3n + 1)
Simplifying this equation, we get:
20 = (n/2)(3n + 5)
40 = n(3n + 5)
Expanding the right side, we get:
40 = 3n^2 + 5n
Rearranging and setting equal to zero, we get:
3n^2 + 5n - 40 = 0
We can solve this quadratic equation using the quadratic formula:
n = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 3, b = 5, and c = -40. Plugging in these values, we get:
n = (-5 ± sqrt(5^2 - 4(3)(-40))) / 2(3)
n = (-5 ± sqrt(625)) / 6
n = (-5 ± 25) / 6
We get two solutions:
n = 4 (ignoring the negative solution)
n = -20/3 (extraneous solution)
Therefore, the number of terms in the sequence is 4.
To find the value of n in the given arithmetic sequence, we need to sum up the terms until we reach the desired sum of 20.
The formula for the sum of an arithmetic series is:
Sn = (n/2)(2a + (n-1)d)
Where:
- Sn is the sum of the series
- n is the number of terms
- a is the first term
- d is the common difference between terms
In this case, the first term (a) is 4 and the common difference (d) is 7 - 4 = 3.
We want to find the sum (Sn) that equals 20. So, we can set up the following equation:
20 = (n/2)(2(4) + (n-1)(3))
Simplifying this equation further, we get:
20 = (n/2)(8 + 3n - 3)
20 = (n/2)(5 + 3n)
Multiply both sides by 2 to get rid of the fraction:
40 = n(5 + 3n)
40 = 5n + 3n^2
Rearrange the equation to quadratic form:
3n^2 + 5n - 40 = 0
Factor the quadratic equation:
(3n - 8)(n + 5) = 0
From here, we can solve for n:
3n - 8 = 0 or n + 5 = 0
3n = 8 or n = -5
n = 8/3 or n = -5
Since n represents the number of terms, it cannot be negative. Therefore, the value of n is 8/3.
Hence, the number of terms required to achieve a sum of 20 in the given sequence is 8/3.