Question Number 114208 by aurpeyz last updated on 17/Sep/20

find the greatest coeeficient in the expansion  of (3+4x)^(−5)

Commented byI want to learn more last updated on 17/Sep/20

Please how sir

Commented bymr W last updated on 17/Sep/20

no maximum, no minimum!

Commented byaurpeyz last updated on 18/Sep/20

why?

Commented bymr W last updated on 18/Sep/20

(3+4x)^(−5)   =3^(−5) ×(1+(4/3)x)^(−5)   =3^(−5) Σ_(k=0) ^∞ (−(4/3))^k C_4 ^(k+4) x^k   with increasing k, ((4/3))^k is increasing  and C_4 ^(k+4)  is aslo increasing. therefore  there is no max/min for 3^(−5) (−(4/3))^k C_4 ^(k+4) ,  it →+∞ or −∞.

Commented byaurpeyz last updated on 18/Sep/20

i think there is more to this topic than i know

Commented bymr W last updated on 18/Sep/20

with that what you know you can  solve alot of questions, such as this  one. just apply!

Commented byaurpeyz last updated on 18/Sep/20

thanks alot Sir

Commented byaurpeyz last updated on 18/Sep/20

when i try to find the greatest coeeficient  using Σ_(r=0) ^∞ C_r ^(5+r−1) ((4/3))^r (−1)^(r )  I arrived at a negative   value for r. I think it is a proof that there is no   greatest coefficient.

Commented bymr W last updated on 18/Sep/20

no proof needed. it′s clear:  C_4 ^(4+r) ((4/3))^r  is always increasing and  (−1)^(r ) =1 for even r and −1 for odd r.  so there is neither maximum nor  minimum coefficient.

Commented bymr W last updated on 18/Sep/20

it′s something like  1,−2,3,−4,5,−6,....  it has no maximum and no minimum.

Commented byaurpeyz last updated on 18/Sep/20

anytime i have a question. how can i know  without proof that there is no max or min.  did you determine it by ((4/3))^r  or the nature  of the question being (3+4x)^(−5) .    i have not seen any textbook that deals with  greatest coefficient of question like that.   i thought it is because of the positive sign (+)   between the binomial.    what if we had a question with ((3/4))^x  ?

Commented bymr W last updated on 18/Sep/20

(−1)^r C_4 ^(4+r) ((4/3))^r has no max and no min.  (−1)^r C_4 ^(4+r) ((3/4))^r  has max and min.

Commented byaurpeyz last updated on 18/Sep/20

thank you Sir. i will attempt it

Commented byaurpeyz last updated on 18/Sep/20

lets say i have (4+3x)^(−5)   (4+3x)^(−5) =4^(−5) [(−1)^r Σ_(r=0) ^∞ C_r ^(4+r) ((3/4))^r ]  (−1)^r C_r ^(4+r) ((3/4))^r ≥ (−1)^(r+1) C_(r+1) ^(5+r) ((3/4))  C_4 ^(4+r) ≥(−1)C_4 ^(5+r) ((3/4))^1   r=((−19)/7).   the negativity gives me issue.    is it that any question like (4+3x)^(−5)  [with   positive sign in between binomial of negative  powers] usually dont have max or min?

Commented bymr W last updated on 18/Sep/20

to treat the positive coefficients  you should only see the even terms,  i.e. r=2k  ......

Commented byaurpeyz last updated on 19/Sep/20

please shed little light on this. r=2k? how  is this done?

Commented bymr W last updated on 19/Sep/20

even terms (r=2k) have positive   coefficients, odd terms (r=2k+1)  have negative coefficients. just think!

Commented byaurpeyz last updated on 20/Sep/20

okay