The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.

Step 1: Define the terms of an AP

Let $a$ be the first term and $d$ be the common difference of the AP. The $n$ -th term of an AP is given by the formula $a_n = a + (n-1)d$. We will use this to express the third and seventh terms.

Step 2: Express the given conditions using the AP formula

We write the third term as $a_3 = a + 2d$ and the seventh term as $a_7 = a + 6d$. The problem states that their sum is 6 and their product is 8. We will form equations based on these conditions.

Step 3: Formulate equations from the given conditions

From the given information, we form two equations. Equation 1 represents the sum of the third and seventh terms, and Equation 2 represents their product. We now have a system of two equations with two variables, $a$ and $d$.

Step 4: Simplify Equation 1

Simplifying Equation 1, we combine like terms: $a + 2d + a + 6d = 2a + 8d = 6$. Dividing by 2, we get $a + 4d = 3$. This simplified equation will be easier to use for substitution.

Step 5: Express 'a' in terms of 'd' from Equation 3

From Equation 3, we can express $a$ in terms of $d$. This allows us to substitute $a$ into Equation 2, thereby reducing it to a single variable equation in $d$.

Step 6: Substitute 'a' into Equation 2

Substitute $a = 3 - 4d$ into Equation 2 $(a + 2d)(a + 6d) = 8$. This step transforms Equation 2 into an equation solely involving $d$, which we can then solve.

Step 7: Simplify and solve for 'd'

Simplify the substituted equation using the difference of squares formula, $(x-y)(x+y) = x^2 - y^2$. This leads to a quadratic equation in $d$, for which we find two possible values for $d$: $1/2$ and $-1/2$.

Step 8: Case 1: Calculate 'a' when $d = 1/2$

Using Equation 3 ($a = 3 - 4d$), we substitute $d = 1/2$ to find the corresponding value for $a$. In this case, $a = 1$. This gives us one possible AP.

Step 9: Calculate sum of first 16 terms for Case 1

The sum of the first $n$ terms of an AP is given by the formula $S_n = \frac{n}{2}[2a + (n-1)d]$. For $n=16$, $a=1$, and $d=1/2$, the sum $S_{16}$ is calculated as 76.

Step 10: Case 2: Calculate 'a' when $d = -1/2$

Now we consider the second possible value for $d$. Substituting $d = -1/2$ into Equation 3 ($a = 3 - 4d$), we find $a = 5$. This gives us another possible AP.

Step 11: Calculate sum of first 16 terms for Case 2

Again, using the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$, with $n=16$, $a=5$, and $d=-1/2$, we find $S_{16} = 20$. Both values of $d$ lead to valid APs.

More from this chapter

• find the number of terms in the AP 7 13 19......205
• is 78 atom of the AP 5 11 17 23
• which time of Ap 3,8,13,18 is 78
• the sum of the first 15 terms of an ap is 450 and its first term is 10 find its common difference
• if the (p)th term of an ap is (q) and the (q)th term is (p) then its (p+q)th term is
• if the teeth term of an ap is Q and the cute term is P then its P + Q to term is
• the fifth time of an ap is 18 and 11th term is 42 find first term
• if the sum of the first n terms of an ap is S sem is equal to 3 and square + 5 and find its common difference

All Arithmetic Progressions solutions →