Starting with t1 equal to 2, the recursive rule gives the sequence 2, 4, 10, 28, 82, 244, 730. Therefore, 730 occurs at the seventh position. Hence, 730 is the seventh term of the sequence.
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The first term is 5 and the common ratio is 5. Using the GP formula, tn equals a multiplied by r raised to n minus 1. Therefore, tn equals 5 raised to n. The 10th term is 9765625.
The 8th term is 192 and the common ratio is 2. From the 8th term to the 12th term there are four steps. Therefore, t12 equals 192 multiplied by 2 four times, giving 3072.
The marble counts represent the sum of the first 25 natural numbers. Using the sum formula Sn = n(n + 1) / 2 with n = 25, the total calculation is 25 x 26 / 2, which gives 325 marbles.
With initial salary of 5,00,000 and annual increment of 20,000, his earnings form an AP. Solving 7,00,000 = 5,00,000 + (n – 1) x 20,000 gives n = 11, meaning his income reaches 7,00,000 after 10 years.