If $t_{n}$ , represents $n^{t h}$ term of an A.P., $t_{2} + t_{5} - t_{3} = 10$ and $t_{2} + t_{9} = 17$ , then find its first term and its common difference.

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Solution

$t_{2} + t_{5} - t_{3} = 10$
Let the first term be $a$ and common difference $= d$
$\Rightarrow t_{n} = a + (n - 1) d$
$t_{2} + t_{5} - t_{3} = 10$
$(a + d) + (a + 4 d) - (a + 2 d) = 10$
$a + 3 d = 10............. (1)$
also,
$\Rightarrow t_{2} + t_{9} = 17$
$a + d + a + 8 d = 17$
$2 a + 9 d = 17............ (2)$

Now by multiply eq. (1) by 2 and subtract eq. (2) from it we got:
$\Rightarrow - 3 d = 3$
$d = - 1$
From eq. (1): $a - 3 = 10$
$\Rightarrow a = 13$

$∴$ First term $= 13$ Common difference $= - 1.$
Hence, the answer is $13, - 1.$

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If tn, represents nth term of an A.P., t2+t5−t3=10 and t2+t9=17, then find its first term and its common difference.

If $t_{n}$ , represents $n^{t h}$ term of an A.P., $t_{2} + t_{5} - t_{3} = 10$ and $t_{2} + t_{9} = 17$ , then find its first term and its common difference.