What is the 12th term from the end of the arithmetic progressions 3 8 13 253?

The 12th term from the end of the arithmetic progression 3, 8, 13, ..., 253 is 198.

Understanding Arithmetic Progressions

An arithmetic progression (AP) is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference. In the given sequence, 3, 8, 13, ..., 253:

• The first term (a) is 3.
• The common difference (d) is 8 - 3 = 5.

Finding the nth Term from the End

To find the nth term from the end of an AP, you can reverse the sequence and treat it as a new AP. The last term of the original AP becomes the first term of this reversed AP.

• Let 'l' be the last term of the original AP.
• The common difference of the reversed AP will be '-d'.

The formula to find the nth term of an AP is:

• a_n = a + (n-1) * d
• Where a is the first term and d is the common difference.

For the reversed AP:

• a_n = l - (n -1) *d

Calculation

In this question:

• l = 253, the last term.
• n = 12, we need the 12th term from the end.
• d = 5, the common difference of the original AP.

So, the 12th term from the end can be calculated by:

a₁₂ = 253 - (12 - 1) 5
a₁₂ = 253 - (11 5)
a₁₂ = 253 - 55
a₁₂ = 198

Conclusion

Therefore, the 12th term from the end of the arithmetic progression 3, 8, 13, ..., 253 is 198, as confirmed by the reference provided.