Question

The 2nd term of an arithmetic progression is (1/2) and the sum of the first 14th terms is -70. Find

Answer 1

Answer:

The common difference is -1.

The last term (the 14th term) is -11.5.

Step-by-step explanation:

In an arithmetic sequence, the second term is 0.5 and the sum of the first 14 terms is -70.

We want to determine the: (a) common difference and (b) the last term.

We can write an explicit formula to represent the sequence. An arithmetic sequence can be modeled by the formula:

$\displaystyle x_n=a+d(n-1)$

Where a is the initial term, d is the common difference, and n represents the nth term.

Since the second term is 0.5:

$x_2=0.5=a+d(2-1)$

Simplify:

$x_2=0.5=a+d$

The sum of an arithmetic sequence is given by the formula:

$\displaystyle S=\frac{k}{2}\left(a+x_k\right)$

Where k is the number of terms and x_k is the last term.

Since the sum of the first 14 terms is -70, S = -70 and k = 14:

Using our explicit formula, the last term is:

$x_{14}=a+d(14-1)=a+13d$

Substitute:

$\displaystyle -70=\frac{14}{2}(a+(a+13d))$

Simplifiy:

$-10=2a+13d$

Rewrite the equation for the second term:

$a=0.5-d$

Substitute:

$-10=2(0.5-d)+13d$

Simplify:

$-10=1-2d+13d$

Solve for d:

$d=-1$

Hence, our common difference is -1.

Solve for a, the initial term:

$a=0.5-(-1)=1.5$

So, our explicit formula is now:

$x_n=1.5-1(n-1)=1.5-n+1=2.5-n$

So, the last term (which is 14) is:

$x_{14}=2.5-(14)=-11.5$