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.
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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.
The two-digit numbers divisible by 3 range from 12 to 99, forming an AP with common difference 3. There are 30 such numbers and applying the arithmetic sum formula gives a total sum of 1665.