Sum
If `t_n` represents nth term of an A.P `t_2 + t_5 - t_3 = 10` and `t_2 + t_9 = 17`. Find its first term and its common difference.
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Solution
Let the first term of an A.P be a and the common difference be d.
The general term of an A.P is given by `t_n = a + (n - 1)d`
Now `t_1 + t_5 - t_3 = 10`
=> (a + d) + (a + 4d) - (a + 2d) = 10
`=> a + d + a + 4d - a - 2d = 10`
=> a + 3d = 10 ...(i)
Also `t_2 + t_9 = 17`
=> (a + d) + (a + 8d) = 17
=> 2a + 9d = 17 ....(ii)
Multiplying equation (i) by 2 we get
2a + 6d = 20 ....(iii)
Subtracting (ii) from (iii) we get
-3d= 3
=> d = -1
Substituting value of d in (i) we get
a + 3(-1) = 10
=> a - 3 = 10
=> a = 13
Hence a = 13 and d = -1
Concept: Arithmetic Progression - Finding Sum of Their First ‘N’ Terms.
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