Answer :
Final answer:
The number of terms in the arithmetic progression (AP) with the sum of 210 is 10.
Explanation:
To find the number of terms in the arithmetic progression (AP) with the sum of 210, we can use the formula for the sum of an AP. The formula is S = (n/2)(2a + (n-1)d)
where
S is the sum
a is the first term
d is the common difference
n is the number of terms.
In this case, we have S = 210, a = 3, and d = 4. Substituting these values into the formula, we get:
210 = (n/2)(2(3) + (n-1)(4))
210 = (n/2)(6 + 4(n-1))
210 = (n/2)(6 + 4n - 4)
210 = (n/2)(4n + 2)
210 = 2n^2 + n
2n^2 + n - 210 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring is not possible in this case, so we'll use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
Using a = 2, b = 1, and c = -210, we get:
n = (-1 ± √(1^2 - 4(2)(-210))) / (2(2))
n = (-1 ± √(1 + 1680)) / 4
n = (-1 ± √1681) / 4
n = (-1 ± 41) / 4
n = (-1 + 41) / 4 = 40 / 4 = 10
So therefore, the number of terms in the AP is 10.
Learn more about arithmetic progression here:
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