Question

second term of hp 3/14 and fifth term is 1/10 find the sum of it's 6th and 7th terms . please do it step by step
​

Answer 1

$given \: 2nd \: term \: of \: an \: hp \: = \frac{3}{14} \\ \\ - > \frac{1}{a + d} = \frac{1}{14 \div 3} - eq(1) \\ \\ 5th \: term \: of \: an \: hp \: = \frac{1}{10} \\ \\ - > \frac{1}{a + 4d} = \frac{1}{10} - eq(2) \\ \\ subtracting \: eq(1) \: from \: eq(2) \\ \\ - > a +4 d = 10 \\ \: \: \: \: \: \: \: \: \: \: \: a + d = \frac{14}{3} \\( - ) \: \: \: \: \: \: \: ( - ) \\ - - - - - - - - - \\ - > 3d \: = \frac{16}{3} \\ - > d = \frac{16}{9} \\ \\ from \: eq(1) \: \: a = \frac{14}{3} - \frac{16}{9} = \frac{42 - 16}{9 } \\ \\ = \frac{26}{9} \\ \\ 6th \: term \: of \: an \: hp \: = a + 5d = \frac{106}{9} \\ \\ 7th \: term \: of \: an \: hp \: = a + 6d = \frac{122}{9} \\ \\ sum = \frac{106}{9} + \frac{122}{9} = \frac{76}{3}$