To determine the correct values for
a and
r, we need to use the properties of infinite geometric series.
The sum of an infinite geometric series can be expressed as:
S=‌We are given that the sum of the infinite series is 4 :
‌=4The second term of the series can be calculated as:
ar We are given that the second term is
‌ :
ar=‌Now, we have two equations:
‌‌=4‌ar=‌From equation (2), we can express
a in terms of
r :
a=‌Substitute this into equation (1):
‌=4Simplify and solve for
r :
‌‌=4‌3=16r(1−r)‌3=16r−16r2‌16r2−16r+3=0 Solving this quadratic equation for
r, we use the quadratic formula:
r=‌Here,
a=16,b=−16, and
c=3 :
‌r=‌| −(−16)±√(−16)2−4⋅16⋅3 |
| 2â‹…16 |
‌r=‌‌r=‌‌r=‌‌r=‌‌‌‌ or ‌‌‌r=‌‌r=‌‌‌‌ or ‌‌‌r=‌ Now, using these values for
r, we find the corresponding
a :
For
r=‌ :
a=‌=1For
r=‌ :
a=‌=3The correct pairs of values are:
a=1‌r=‌a=3‌r=‌ So the correct option from the given choices is:
Option B:
a=3‌‌r=‌