Final answer:
In an arithmetic sequence, given the information provided (t4 = 10 and tg = 4), we can find the first term and the common difference of the sequence using the formula tn = a + (n - 1)d. By solving the equations, we can find the values of a = -2 and d = 4. Using these values, we can find t10 = 38 and s27 = 540.
Step-by-step explanation:
In an arithmetic sequence, the formula to find the nth term is given by: tn = a + (n - 1)d, where tn is the nth term, a is the first term, n is the position of the term, and d is the common difference. Given t4 = 10 and tg = 4, we can find the first term and the common difference using these values.
Using the formula, we can substitute the known values to find the first term: 10 = a + (4 - 1)d. Simplifying, we get 10 = a + 3d. The other equation is tg = 4, which can be rewritten as a + (g - 1)d = 4 with g=4.
By solving these two equations simultaneously, we can find the values of a and d. Substituting the value of a from the first equation into the second equation, we get (10 - 3d) + (4 - 1)d = 4. Simplifying, we get 13 - 2d = 4, which leads to d = 4. Substituting this value of d back into the first equation, we can find the value of a: 10 = a + 3(4), which gives a = -2.
Now that we have the values of a and d, we can use the formula tn = a + (n - 1)d to find t10 and s27.
Substituting the values into the formula, we get t10 = -2 + (10 - 1) * 4 = 38 and s27 = (27/2)(-2 + 38) = 540.