Question Number 114879 by bobhans last updated on 21/Sep/20

If the product of the matrices    (((1   1)),((0   1)) ) (((1    2)),((0    1)) ) (((1     3)),((0     1)) )... (((1     k)),((0     1)) )= (((1    378)),((0        1)) )  then k =

Answered by john santu last updated on 21/Sep/20

(i) (((1   1)),((0   1)) ) (((1    2)),((0    1)) )= (((1   3)),((0   1)) )  (ii) (((1    1)),((0    1)) ) (((1     2)),((0     1)) ) (((1   3)),((0   1)) )= (((1    3)),((0   1)) ) (((1  3)),((0   1)) )= (((1   6)),((0   1)) )  (iii) (((1    6)),((0   1)) ) (((1    4)),((0    1)) )= (((1     10)),((0       1)) )  (iv) (((1    10)),((0      1)) ) (((1     5)),((0     1)) )= (((1     15)),((0       1)) )  therefore    (((1   1)),((0   1)) ) (((1    2)),((0    1)) ) (((1    3)),((0    1)) )... (((1     k)),((0    1)) )= (((1    378)),((0       1)) )   (((1     1+2+3+...+k)),((0                   1)) )= (((1     378)),((0        1)) )  ⇔ 1+2+3+...+k = 378       ((k(k+1))/2) = 378 ⇒k^2 +k−756=0      (k+27)(k−28) = 0     ⇔ k = 28 .