The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.
(i) Find the first term of the progression and the common difference.
The nth term of the progression is 46.
(ii) Find the value of n.
Solution
(i) The formula for the nth term of an arithmetic progression is given by:
\(a_n = a + (n-1)d\)
For the ninth term, \(a + 8d = 22\).
The sum of the first 4 terms is given by:
\(S_4 = \frac{4}{2}(2a + 3d) = 49\)
\(2a + 3d = 24.5\)
We have the system of equations:
1. \(a + 8d = 22\)
2. \(2a + 3d = 24.5\)
Solving these equations simultaneously, we find:
\(d = 1.5\)
\(a = 10\)
(ii) The nth term is given by:
\(a + (n-1)d = 46\)
Substituting \(a = 10\) and \(d = 1.5\):
\(10 + (n-1) \times 1.5 = 46\)
\(1.5(n-1) = 36\)
\(n-1 = 24\)
\(n = 25\)
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