S=1+2(1
5
)+3
(1
5
)
2+4
(1
5
)
3…. If
S=a
b
,
where a and
b are coprime positive integers, what is the
value of a+b?
To find the value of S, let's simplify the given expression step by step:
S = 1 + 2(1/5) + 3(1/5)^2 + 4(1/5)^3
Step 1: Evaluate the exponentials:
S = 1 + 2/5 + 3(1/5^2) + 4(1/5^3)
Step 2: Simplify the fractions:
S = 1 + 2/5 + 3/25 + 4/125
Step 3: Find a common denominator:
S = (125/125) + (50/125) + (15/125) + (4/125)
Step 4: Add the fractions:
S = (125 + 50 + 15 + 4) / 125
Step 5: Combine the numerator:
S = 194/125
Since S is in the form of a fraction (194/125), a is 194 and b is 125. To find the value of a + b:
a + b = 194 + 125 = 319
Therefore, the value of a + b is 319.