The second term of a linear sequence is 15 and the 5th term is 21.Find the first term,the common difference and the sum of the first 16 term
just use your definitions:
a + d = 15
a + 4d = 21
subtract them
3d = 6
d = 2
mentally from a+d = 15, a = 13
sum(n) = (n/2)(2a + (n-1)d)
= 8(26 + 15(2))
= ...
To find the first term, common difference, and sum of the first 16 terms in a linear sequence, we can use the formulas for the nth term and the sum of an arithmetic series.
Let's start with finding the common difference (d):
The nth term of a linear sequence can be calculated using the formula:
an = a1 + (n - 1) * d
Where:
an is the nth term,
a1 is the first term,
n is the term position in the sequence, and
d is the common difference.
By substituting the given information, we can solve for the common difference:
15 = a1 + (2 - 1) * d (since the second term is 15)
21 = a1 + (5 - 1) * d (since the fifth term is 21)
From the first equation, we have:
15 = a1 + d
And from the second equation, we have:
21 = a1 + 4d (multiply 4 by d)
Now, we can create an equation by subtracting the first equation from the second equation:
21 - 15 = (a1 + 4d) - (a1 + d)
6 = 3d
Therefore, the common difference (d) is 2.
Now, let's find the first term (a1):
Using the first equation from above:
15 = a1 + d
Substituting the value of d (2) into the equation:
15 = a1 + 2
By rearranging the equation, we find:
a1 = 15 - 2
a1 = 13
Thus, the first term (a1) is 13.
Now, let's find the sum of the first 16 terms (Sn):
The sum of the first n terms in an arithmetic series can be calculated using the formula:
Sn = (n/2) * (2a1 + (n-1)d)
Substituting the given values:
n = 16 (since we want to find the sum of the first 16 terms),
d = 2,
and a1 = 13,
We can calculate the sum as follows:
Sn = (16/2) * (2 * 13 + (16 - 1) * 2)
= 8 * (26 + 15 * 2)
= 8 * (26 + 30)
= 8 * 56
= 448
Therefore, the sum of the first 16 terms (Sn) is 448.