Question

The sum of the first N terms of the arithmetic progression 4,10,16... equals the 64th term of the arithmetic progression 21,42,63,... find N?​

Answer 1

Answer 21

The solotion to this task

Answer 2

To find the value of N, which represents the number of terms in the arithmetic progression 4, 10, 16, ..., that add up to the 64th term of the arithmetic progression 21, 42, 63, ..., we need to follow these steps:

Step 1: Find the common difference (d) for both arithmetic progressions.

The common difference (d) for an arithmetic progression is the constant value added to each term to get the next term. For the first arithmetic progression (4, 10, 16, ...), we can observe that each term increases by 6:

4 + 6 = 10, 10 + 6 = 16, ...

So, the common difference (d) for the first arithmetic progression is 6.

Similarly, for the second arithmetic progression (21, 42, 63, ...), each term increases by 21:

21 + 21 = 42, 42 + 21 = 63, ...

So, the common difference (d) for the second arithmetic progression is 21.

Step 2: Find the value of the 64th term in the second arithmetic progression.

The formula to find the nth term of an arithmetic progression is given by:

nth term = a + (n - 1) * d,

where a is the first term and d is the common difference.

For the second arithmetic progression, a = 21 and d = 21, and we want to find the 64th term (n = 64):

64th term = 21 + (64 - 1) * 21 64th term = 21 + 63 * 21 64th term = 21 + 1323 64th term = 1344.

Step 3: Find the number of terms (N) in the first arithmetic progression.

Now, we want to find the sum of the first N terms in the first arithmetic progression (4, 10, 16, ...), which is equal to the 64th term (1344) of the second arithmetic progression.

The formula to find the sum of the first N terms of an arithmetic progression is given by:

Sum of N terms = (N/2) * [2a + (N - 1) * d],

where a is the first term and d is the common difference.

For the first arithmetic progression, a = 4 and d = 6:

N/2 * [2 * 4 + (N - 1) * 6] = 1344.

Simplify the equation:

N * [8 + 6(N - 1)] = 2688, N * [8 + 6N - 6] = 2688, N * (6N + 2) = 2688, 6N^2 + 2N - 2688 = 0.

Step 4: Solve for N.

Now, we need to solve the quadratic equation for N:

6N^2 + 2N - 2688 = 0.

This equation can be factored as follows:

(N + 42)(6N - 64) = 0.

Setting each factor to zero and solving for N:

1. N + 42 = 0 N = -42 (We discard this negative value as the number of terms cannot be negative)

2. 6N - 64 = 0 6N = 64 N = 64 / 6 N = 10.67 (approx).

Since the number of terms (N) must be a positive integer, we round down the approximate value to the nearest whole number:

N ≈ 10.

So, the number of terms (N) in the first arithmetic progression is 10.