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11 being the first term in the above sequence, and, each successive t
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19 May 2025, 09:57
19 May 2025, 09:57
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$$
\(11,15,19,23 \ldots \ldots, 91\)
$$
11 being the first term in the above sequence, and, each successive term is obtained by adding 4 to the previous term. What is the number of terms in the above sequence?
\(11,15,19,23 \ldots \ldots, 91\)
$$
11 being the first term in the above sequence, and, each successive term is obtained by adding 4 to the previous term. What is the number of terms in the above sequence?
Show: ::
21
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Re: 11 being the first term in the above sequence, and, each successive t
[#permalink]
21 May 2025, 04:00
21 May 2025, 04:00
Expert Reply
We are given the arithmetic sequence:
$$
\(11,15,19,23, \ldots, 91\)
$$
- First term \(( $a_1$ ): 11\)
- Common difference $\((d): 4\)$ (since each term increases by 4 )
- Last term \($\left(a_n\right)$ : 91\)
We need to find the number of terms ( $n$ ) in this sequence.
Step 1: Use the Arithmetic Sequence Formula
The general formula for the $n$-th term of an arithmetic sequence is:
$$
\(a_n=a_1+(n-1) \times d\)
$$
Step 2: Plug in the Known Values
Substitute $\(a_n=91, a_1=11\)$, and $\(d=4\)$ into the formula:
$$
\(91=11+(n-1) \times 4\)
$$
Step 3: Solve for $n$
1. Subtract 11 from both sides:
$$
\(\begin{gathered}
91-11=(n-1) \times 4 \\
80=(n-1) \times 4
\end{gathered}\)
$$
2. Divide both sides by 4:
$$
\(\begin{aligned}
& \frac{80}{4}=n-1 \\
& 20=n-1
\end{aligned}\)
$$
3. Add 1 to both sides:
$$
\(n=21\)
$$
Verification
To ensure correctness, let's verify the 21st term:
$$
\(a_{21}=11+(21-1) \times 4=11+80=91\)
$$
This matches the given last term.
Final Answer
The number of terms in the sequence is:
21
$$
\(11,15,19,23, \ldots, 91\)
$$
- First term \(( $a_1$ ): 11\)
- Common difference $\((d): 4\)$ (since each term increases by 4 )
- Last term \($\left(a_n\right)$ : 91\)
We need to find the number of terms ( $n$ ) in this sequence.
Step 1: Use the Arithmetic Sequence Formula
The general formula for the $n$-th term of an arithmetic sequence is:
$$
\(a_n=a_1+(n-1) \times d\)
$$
Step 2: Plug in the Known Values
Substitute $\(a_n=91, a_1=11\)$, and $\(d=4\)$ into the formula:
$$
\(91=11+(n-1) \times 4\)
$$
Step 3: Solve for $n$
1. Subtract 11 from both sides:
$$
\(\begin{gathered}
91-11=(n-1) \times 4 \\
80=(n-1) \times 4
\end{gathered}\)
$$
2. Divide both sides by 4:
$$
\(\begin{aligned}
& \frac{80}{4}=n-1 \\
& 20=n-1
\end{aligned}\)
$$
3. Add 1 to both sides:
$$
\(n=21\)
$$
Verification
To ensure correctness, let's verify the 21st term:
$$
\(a_{21}=11+(21-1) \times 4=11+80=91\)
$$
This matches the given last term.
Final Answer
The number of terms in the sequence is:
21
gmatclubot
Re: 11 being the first term in the above sequence, and, each successive t [#permalink]
21 May 2025, 04:00
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