Sum

If `t_n` represents n^{th }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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