The sum of the first ten terms of a particular arithmetic sequence is four times the sum of the first five terms of the sequence. What is the ratio of the first term to the second term? Express your answer as a common fraction.

Answer: - 1

Step-by-step explanation:

A progression or sequence is an arrangement of numbers in a definite pattern.

Formula for calculating sum of arithmetic sequence is given by;

Sn = n/2{2a + (n-1) d}

n is the number of terms

a is the first term.

d is the common difference

Sum of first term terms will be

S10 = 10/2{2a + (10-1) d}

S10 = 5{2a+9d} = 10a + 45d

Similarly, for first five terms

S5 = 5/2{2a+4d} = 5a+10d

Since the sum of the first ten terms of the sequence is four times the sum of the first five terms, we have

10a+45d=4 (5a+10d)

10a + 45d=20a + 40d

Dividing through by 5

2a+9d = 4a + 10d

2a+d=0

d = - 2a

If the first term is 'a' and common difference is '-2a'

Second term will be;

a+d i. e a + (-2a) = - a

Ratio of first term to second term will be a/-a = - 1