Sequences

Step 1: Let common difference be d.
Step 2: A₇ = 1+7d, Am-₁=1+(m-1)d
Step 3: (1+7d)/(1+(m-1)d)=(5)/(9)
Step 4: Solving gives m=20.

The two letters, the first and the last of the four lettered word can be chosen in ways, as repetition is allowed for consonants.
The two vowels in the middle are distinct so that the number of ways of filling up the two places is .
The no. of different words .

Step 1: Each term is
Step 2:
Step 3:
Step 4:

Step 1: General term:

Step 2: Set power .
Step 3: Coefficient: