Find the sum of all odd numbers of four digit | vedclass

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Question

(A) The four-digit odd numbers divisible by $9$ form an arithmetic progression ($A$.$P$.).The smallest four-digit odd number divisible by $9$ is $1017

Vedclass · Accepted Answer

$2754000$

Vedclass · Answer

(A) The four-digit odd numbers divisible by $9$ form an arithmetic progression ($A$.$P$.).The smallest four-digit odd number divisible by $9$ is $1017$ (since $1008$ is even and $1017$ is odd).The largest four-digit odd number divisible by $9$ is $9999$.Since the numbers must be odd and divisible by $9$,the common difference $d$ between consecutive terms is $18$ (because $9 	imes 2 = 18$).Using the formula for the $n$-th term of an $A$.$P$.,$a_n = a + (n - 1)d$:$9999 = 1017 + (n - 1) 	imes 18$$8982 = (n - 1) 	imes 18$$n - 1 = 8982 / 18 = 499$$n = 500$The sum $S_n$ of an $A$.$P$. is given by $S_n = \frac{n}{2}(a + l)$:$S_{500} = \frac{500}{2}(1017 + 9999)$$S_{500} = 250 	imes 11016 = 2754000$.

Find the sum of all odd numbers of four digits which are divisible by $9.$

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