Progression Problems - easy

Solution:
We know that and . Therefore either or . But 0 cannot be a member of a geometric progression, so the only option left is . Then the arithmetic progression is or => .

Solution:
The sum is .

Solution:
Since is an arithmetic progression, , therefore . By the geometric progression's property, we get
: or . But the arithmetic progression is non-constant, therefore , so a must be -2. .

Solution:
By the property of the geometric progression, we get
. We also know that , or Substituting into the previous equation, we get , or , which means that . Then and their arithmetic mean is .

Solution:
By the mean property of the arithmetic progression, we have , or . The geometric progression becomes . By its mean property, , or .

Solution:
By the mean property of the geometric progression, we have , or . By the mean property of the arithmetic progression, we have , so .