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.

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