Maths / Arithmetic Progression / General Term Of an AP

QUESTION

For an AP, if p times the pth term is equal to q times the qth term, prove that (p+q)th term is 0

EXPLANATION
Explain TypeExplanation Content
Text

$\dpi{120} \fn_jvn \large a_p=a+(p-1)d$

$\dpi{120} \fn_jvn \large a_q=a+(q-1)d$

A.T.Q.

$\dpi{120} \fn_jvn \large p\; X \;a_p= q \;X\;a_q$

$\dpi{120} \fn_jvn \large p(a+(p-1)d)= q(a+(q-1)d)$

$\dpi{120} \fn_jvn \large ap+(p^2-p)d)= aq+(q^2-q)d$

$\dpi{120} \fn_jvn \large ap+(p^2-p)d - aq-(q^2-q)d =0$

$\dpi{120} \fn_jvn \large a(p-q)+(p^2-p-q^2+q)d =0$

$\dpi{120} \fn_jvn \large a(p-q)+(p^2-q^2-p+q)d =0$

$\dpi{120} \fn_jvn \large a(p-q)+((p+q)(p-q)-1(p-q))d =0$

$\dpi{120} \fn_jvn \large (p-q)(a+((p+q)-1)d =0$

$\dpi{120} \fn_jvn \large but \;p-q \;cannot\; be\; zero\; as\; p\;& \;q \;are\; different\; terms$

$\dpi{120} \fn_jvn \large a+((p+q)-1)d =0$

$\dpi{120} \fn_jvn \large a_{p+q} =0$

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