If a_1, a_2, a_3, . . . , a_n, . . . are in A | vedclass

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(C) Let the first term be $a$ and the common difference be $d$.The terms of an $A.P.$ are given by $a_n = a + (n-1)d$.Given the equation: $a_4 - a_7 +

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$13$

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(C) Let the first term be $a$ and the common difference be $d$.The terms of an $A.P.$ are given by $a_n = a + (n-1)d$.Given the equation: $a_4 - a_7 + a_{10} = m$.Substituting the general form:$(a + 3d) - (a + 6d) + (a + 9d) = m$$a + 3d - a - 6d + a + 9d = m$$a + 6d = m$Note that $a_7 = a + (7-1)d = a + 6d$,so $a_7 = m$.The sum of the first $13$ terms $(S_{13})$ is given by the formula:$S_{13} = \frac{13}{2} [2a + (13-1)d]$$S_{13} = \frac{13}{2} [2a + 12d]$$S_{13} = 13(a + 6d)$Since $a + 6d = m$,we have:$S_{13} = 13m$.

If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

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