1, 4, 7, 10, ..., 301
For this sequence, the first sequence is a_1= 1, the common difference is d=4-1= 3, and the last term is a_n= 301. First, let's find the number of terms in the given sequence. We will use the formula for the explicit rule of arithmetic sequences.
a_n= a_1+(n-1) dNow, we will substitute values into the formula and solve if for n.
Next, we will calculate the sum of the given sequence.
1+4+7+...+301
To do this, we will use the formula for finding the sum of an arithmetic series for n=101.
S_()darkorangen=n(a_1+a_()darkorangen)/2
⇓
S_()darkorange101=101(a_1+a_()darkorange101)/2
Finally, we will substitute 1 for a_1 and 301 for a_(101), and then find the sum.
This tells us that the sum of the given arithmetic sequence is 15 251.
b We are asked to find the sum of the following arithmetic sequence.
1, 2, 3, 4, ..., 1000
For this sequence, the first sequence is a_1= 1, the common difference is d=2-1= 1, and the last term is a_n= 1000. First, let's find the number of terms in the given sequence. We will use the formula for the explicit rule of arithmetic sequences.
a_n= a_1+(n-1) dNow, we will substitute values into the formula and solve if for n.
Next, we will calculate the sum of the given sequence.
1+2+3+...+1000
To do this, we will use the formula for finding the sum of an arithmetic series for n=1000.
S_()darkorangen=n(a_1+a_()darkorangen)/2
⇓
S_()darkorange1000=1000(a_1+a_()darkorange1000)/2
Finally, we will substitute 1 for a_1 and 1000 for a_(1000), and then find the sum.
This tells us that the sum of the given arithmetic sequence is 500 500.
c We are asked to find the sum of the following arithmetic sequence.
2, 4, 6, 8, ..., 800
For this sequence, the first sequence is a_1= 2, the common difference is d=4-2= 2, and the last term is a_n= 800. First, let's find the number of terms in the given sequence. We will use the formula for the explicit rule of arithmetic sequences.
a_n= a_1+(n-1) dNow, we will substitute values into the formula and solve if for n.
Next, we will calculate the sum of the given sequence.
2+4+6+...+800
To do this, we will use the formula for finding the sum of an arithmetic series for n=400.
S_()darkorangen=n(a_1+a_()darkorangen)/2
⇓
S_()darkorange400=400(a_1+a_()darkorange400)/2
Finally, we will substitute 2 for a_1 and 800 for a_(400), and then find the sum.
