Question

3+7+11+15+19+...Sn=210n=?

Answer 1

$a_1=3;\ a_2=7\ =\ \textgreater \ d=7-3=4\\ a_n=a_1+d(n-1)= 3+4(n-1)=4n-1\\ s_n=\dfrac{a_1+a_n}{2}*n=\dfrac{3+4n-1}{2}*n=(1+2n)n\\ S_n=210\ =\ \textgreater \ (1+2n)n=210\\ 2n^2+n-210=0,\ n \in N\\ D= 1681;\ =\ \textgreater \ n=10$
Ответ: 10

Answer 2

The given series is an arithmetic sequence with a common difference of 4 between consecutive terms. The general formula for the nth term of an arithmetic sequence is given by:

a_n = a_1 + (n - 1) * d

Where: a_n = nth term a_1 = first term d = common difference n = term number

In this case: a_1 = 3 (the first term) d = 4 (the common difference)

Now, we need to find the sum of the first n terms of the series, which is given by the formula for the sum of an arithmetic sequence:

S_n = (n/2) * (a_1 + a_n)

Substitute the values of a_1 and a_n:

S_n = (n/2) * (3 + (3 + (n - 1) * 4)) S_n = (n/2) * (3 + 3 + 4n - 4) S_n = (n/2) * (6 + 4n)

Now, we want to find the value of Sn when S_n = 210n:

210n = (n/2) * (6 + 4n)

To solve for n, let's first simplify the equation:

210n = 3n + 2n^2

Now, let's put the equation in the standard quadratic form:

2n^2 + 3n - 210n = 0

Next, factor out common terms:

n(2n + 3 - 210) = 0

Now, set each factor to zero and solve for n:

1. n = 0 (Ignore this solution since it doesn't make sense in this context.)
2. 2n + 3 - 210 = 0 2n = 210 - 3 2n = 207 n = 207 / 2 n = 103.5

Since we are dealing with the number of terms, n cannot be a decimal; it must be a positive whole number. So, we can't use n = 103.5.

However, we can check if the sum of the first 103 terms is equal to 210 times the number of terms:

S_103 = (103/2) * (6 + 4 * 103) S_103 = 53 * 418 S_103 = 22154

Now, let's check if 22154 = 210 * 103:

210 * 103 = 21630

Since 22154 is less than 21630, we can conclude that n = 103 is the correct number of terms.

So, the value of 210n is 210 * 103 = 21630.