Answer :
Final Answer:
The 7th term from the last of the arithmetic progression (AP) is 63.
Explanation:
To find the 7th term from the last of the AP, we first need to find the common difference (d). Given that the AP is a three-digit number and divisible by 7, we can determine the common difference by subtracting the first term (105) from the second term (112).
[tex]\[ d = 112 - 105 = 7 \][/tex]
Now, we can find the 7th term from the last by subtracting 7 from the last term (112) six times:
[tex]\[ 112 - (6 \times 7) = 112 - 42 = 70 \][/tex]
Then, since we need the term before the last term, we subtract the common difference (7) once more:
[tex]\[ 70 - 7 = 63 \][/tex]
Therefore, the 7th term from the last of the AP is 63.
Final answer:
To find the 7th term from the last in the given AP where the first term is 105 and common difference is 7, determine the total number of terms, then apply the formula for the nth term from the start, ultimately concluding that the 7th term from the end is 952.
Explanation:
The question asks to find the value of the 7th term from the last in an arithmetic progression (AP). We are given the first term (A) is 105, the common difference (d) is 7, and every term is a three-digit number divisible by 7. To find the nth term of an AP, we use the formula: Tn = A + (n - 1) * d. However, this formula gives us the nth term from the start of the series. To find the nth term from the end, we first need to determine the total number of terms (N) in the AP, which can be done by finding the largest three-digit number divisible by 7.
Since 999 is the largest three-digit number, we divide it by 7 to find the largest multiple of 7 that is a three-digit number:
999 ÷ 7 ≈ 142.71 This means that 142 * 7 = 994 is the last term (TN) of the AP. Now, using the nth term from the start formula, we can find N:
- 994 = 105 + (N - 1) * 7
N = (994 - 105) / 7 + 1
N ≈ 128 - Term = TN-6
Term = 105 + (128 - 6 - 1) * 7
Term = 105 + (121) * 7
Term = 105 + 847
Term = 952
Thus, the 7th term from the last of the AP is 952, which is the final solution to this problem.
