The 4th term of an AP is 11. The sum of the 5th and 7th terms of this AP is 34. Find its common difference

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The 4th term of an AP is 11. The sum of the 5th and 7th terms of this AP is 34. Find its common difference

Solution:

Suppose that a is the first term and d is the common difference of the AP. Then, we have:

${{a}_{4}}=11$

$a+(4-1)d=11$

$a+3d=11\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$

$Now,$

${{a}_{5}}+{{a}_{7}}=34$

$(a+4d)+(a+6d)=34$

$2a+10d=34$

$a+5d=17\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$

$From\,(1)\,and\,(2),we\,get$

$11-3d+5d=17$

$2d=17-11=6$

$d=3$

$Hence,\text{ }the\text{ }common\text{ }difference\text{ }of\text{ }the\text{ }AP\text{ }is\text{ }3$

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