The first and last term of arithmetic expression are in the ratio 3:4. The sum of all terms is 210, and there are 6 terms in total. Compute common difference.
Let's assume that the first term of the arithmetic expression is 3x and the last term is 4x, where x is a common factor.
We know that the sum of an arithmetic series can be calculated using the formula:
Sum = (n/2) * (first term + last term)
Plugging in the given values, we have:
210 = (6/2) * (3x + 4x)
210 = 3 * (7x)
210 = 21x
To find the value of x, we can divide both sides of the equation by 21:
210/21 = x
10 = x
Now that we know the value of x, we can determine the common difference by subtracting the first term from the second term:
4x - 3x = 4*10 - 3*10 = 40 - 30 = 10
Therefore, the common difference is 10.
To find the common difference in an arithmetic sequence, we need to know the first term, the last term, and the total number of terms. In this case, we know that the first and last terms are in the ratio 3:4, the total sum of all terms is 210, and there are 6 terms in total.
Let's start by finding the first and last terms. Let's assume that the first term is 3x and the last term is 4x, where x is the common difference.
We know that the sum of an arithmetic sequence can be calculated using the formula: sum = (n/2)(first term + last term), where n is the total number of terms. In this case, the sum is 210 and there are 6 terms.
So, using the formula, we can write the equation as:
210 = (6/2)(3x + 4x)
Simplifying the equation:
210 = 3(7x)
70 = 7x
x = 10
Therefore, the common difference (x) is 10.
Let the first term be $3a$ and the last term be $4a$, where $a$ is some positive real number. Since there are 6 terms and the sum is 210, then the average term is $\frac{210}{6} = 35$. Since the terms form an arithmetic progression, the average term is also the median term. In particular, $35$ is the third term. Since we know that the first and last terms add to $3a+4a=210$ and that the third term is $35$, we can write \begin{align*}
3a+4a&=210 \\
7a&=210 \\
a&=30.
\end{align*}So the common difference is $4a-3a =\boxed{30}$.