sequence

10a)4th row : 13 , 15 , 17 , 19
5th row : 21 , 23 , 25 , 27 , 29
b i)t1= The ( (1+2+3+...k-1) + 1 ) th odd number
= The k(k-1)/2 + 1 th odd number
= 1 + 2 ( k(k-1)/2 + 1 - 1 )
= k² - k + 1tk = The (1+2+3+...+k) th odd number
= The k(k+1)/2 th odd number
= 1 + 2 ( k(k+1)/2 - 1 )
= k² + k - 1
b ii)Sum of all the integers in the kth row= (t1 + tk) k / 2= (k² - k + 1 + k² + k - 1) k / 2= k³
c)1³ + 2³ + 3³ + ... + k³= 1 + (3+5) + (7+9+11) + ... + (t1 + tk) k / 2= 1 + 3 + 5 + 7 + 9 + 11 + ... + t1 + t2 + t3 + ... + tk ..... (1+2+3+...+k) terms= (1 + tk) (1+2+3+...+k) / 2= (1 + k² + k - 1) (k(k+1)/2) / 2= k(k+1) (k(k+1)/4)= (1/4) k² (k + 1)²