Three numbers are in an Arithmetic progression with a common difference equal to ¼.
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Three numbers are in an Arithmetic progression with a common difference equal to ¼. The first term of an Arithmetic progression is equal to the sum of the second and third term. What will be the ratio of first and third term?
Show answer & explanation
Let the first term be a. The three terms in AP are: a, a+1/4, a+1/2. Given that a = (a+1/4) + (a+1/2), we get a = 2a + 3/4, so a = -3/4. The third term is -3/4 + 1/2 = -1/4. Therefore, the ratio is (-3/4):(-1/4) = 3:1.
Step-by-step Derivation:
Step 1: Let the first term be 'a' with common difference d = 1/4.
Step 2: The three terms are:
- First term: a
- Second term: a + 1/4
- Third term: a + 2(1/4) = a + 1/2
Step 3: Apply the given condition: first term = second term + third term
a = (a + 1/4) + (a + 1/2)
a = 2a + 3/4
a - 2a = 3/4
-a = 3/4
a = -3/4
Step 4: Calculate the third term:
Third term = a + 1/2 = -3/4 + 1/2 = -3/4 + 2/4 = -1/4
Step 5: Find the ratio:
Ratio = (First term) : (Third term) = (-3/4) : (-1/4) = 3 : 1
Verification:
- First term: -3/4
- Second term: -3/4 + 1/4 = -1/2
- Third term: -1/4
- Check: -3/4 = -1/2 + (-1/4) = -2/4 - 1/4 = -3/4 ✓