Final answer:
In an Arithmetic Progression (AP), the first term and common difference can be found using equations with known terms. In a Geometric Progression (GP), the first term and common ratio can be found using equations with known terms. The sum of the first n terms in an AP or GP can be found using the appropriate formula.
Step-by-step explanation:
A. To find the first term and common difference of an arithmetic progression (AP), we can use the formula:
an = a1 + (n-1)d
Given that the 4th term is 20 and the 8th term is 60, we can substitute these values to get:
a4 = a1 + 3d = 20
a8 = a1 + 7d = 60
Solving these equations simultaneously, we find that the first term (a1) is 5 and the common difference (d) is 15.
To find the 15th term, we can use the same formula:
a15 = a1 + (15-1)d
Substituting the values we found (a1 = 5, d = 15), we get:
a15 = 5 + 14(15) = 215
B. To find the first term and common ratio of a geometric progression (GP), we can use the formula:
an = a1r^(n-1)
Given that the 4th term is 250 and the 7th term is 31,250, we can substitute these values to get:
a7 = a1r^6 = 31,250
Solving these equations simultaneously, we find that the first term (a1) is 10 and the common ratio (r) is 5.
To find the sum of the first 10 terms, we can use the formula:
Sn = a1(1 - r^n) / (1 - r)
Substituting the values we found (a1 = 10, r = 5, n = 10), we get:
S10 = 10(1 - 5^10) / (1 - 5) = 9,765
C. To find the first term, common difference, and sum of the first 8 terms of an arithmetic progression (AP), we can use the given information:
Given that the 8th term is twice the 4th term and the 20th term is 40, we can write the equations:
a8 = 2a4
a20 = 40
Using the formula for the nth term of an AP an = a1 + (n-1)d, we can substitute the values:
a8 = a1 + 7d = 2(a1 + 3d)
a20 = a1 + 19d = 40