For three consecutive months, a person deposited some amount of money on the first day of ech montg in a small saving fund. These three successive amounts in the deposiy, the total value of which is ksh. 65,form a GP if the 2 extreme amounts be multiplied each by 3 abd the mean by 5, the product forms an AP. Find the amounts in the first and secont deposit
To solve this problem, let's break it down step by step.
Step 1: Let's assume that the three successive amounts deposited in the saving fund are a, ar, and ar^2, where a represents the first deposit, r represents the common ratio, and ar^2 represents the third deposit.
Step 2: We know that the sum of these three amounts is equal to 65:
a + ar + ar^2 = 65
Step 3: We also know that the two extreme amounts, a and ar^2, when multiplied by 3, and the mean, ar, when multiplied by 5, form an arithmetic progression (AP). This means:
3a, 5ar, and 3ar^2 form an AP
Step 4: Using the formula for an arithmetic progression, we can write:
5ar - 3a = 3ar^2 - 5ar
Step 5: Simplifying the equation, we get:
5ar - 3a = (3r - 5ar)(r)
Step 6: Expanding and simplifying further, we obtain:
5ar - 3a = 3r^2 - 5ar^2
Step 7: Rearranging the terms, we get:
5ar^2 - 5ar - 3r^2 + 3a = 0
Step 8: Now, let's solve the simultaneous equations formed by steps 2 and 7. We have:
a + ar + ar^2 = 65 --(1)
5ar^2 - 5ar - 3r^2 + 3a = 0 --(2)
Step 9: We can substitute the value of a from equation (1) into equation (2), and then solve for r.
Step 10: After finding the value of r, substitute it back into equation (1) to solve for a.
By following these steps, you should be able to find the amounts in the first and second deposits.